hep-lat/9409003 6 Sep 1994 algorithm malization group two parameter theory Monte Carlo pivot algorithm Karp Luby universal KEY Formerly with the Department of Physics New York University WORDS amplitude Critical for and Two Self avoiding ratio Universal Department of Self Avoiding Internet New York NY Debt Exp onents North second New York NY and Internet September World Financial Center Department Internet and Equity New York University York Ontario M J P walk Washington Place York University MADRAS NEXUS YORKU CA Alan D Sokal Merril l Lynch virial Neal Madras Mathematics and Statistics CANADA Three Dimensional Keele Street Bin Li Amplitude SOKAL NYU EDU BLI ML COM p olymer co e cient Markets Group of Physics Hyp erscaling Walks USA critical USA interpenetration Ratios exp onent hyperscaling ratio renor
Abstract
We make a high precision Monte Carlo study of two and three dimensional
self avoiding walks SAWs of length up to steps using the pivot algo
rithm and the Karp Luby algorithm We study the critical exp onents and
as well as several universal amplitude ratios in particular we make
4
an extremely sensitive test of the hyperscaling relation d In two
4
dimensions we con rm the predicted exp onent and the hyperscal
2 2
ing relation we estimate the universal ratios hR i hR i
g e
2 2
hR i hR i and con dence
m e
limits In three dimensions we estimate with a correction
to scaling exp onent sub jective con dence limits This
1
value for agrees excellently with the eld theoretic renormalization group
prediction but there is some discrepancy for Earlier Monte Carlo esti
1
mates of which were are now seen to b e biased by corrections to
2 2
scaling We estimate the universal ratios hR i hR i and
g e
since hyperscaling holds The approach to is
from ab ove contrary to the prediction of the two parameter renormalization
group theory We critically reexamine this theory and explain where the error
lies
Contents
Introduction
The Problem of Hyp erscaling
Which Quantities are Universal
Plan of this Paper
Background and Notation
The Self Avoiding Walk SAW A Review
The Pivot Algorithm A Review
Algorithms for Counting Overlaps
Generalities
Notation
Deterministic Algorithms
Monte Carlo Algorithms Generalities
Hit or Miss Monte Carlo Algorithm
Barrett Algorithm Theory
Karp Luby Algorithm Theory
Scaling Theory
Barrett and Karp Luby Algorithms Numerical Results
Numerical Results
Two Dimensions
Three Dimensions
Discussion
Comparison with Previous Numerical Studies
Two Dimensions
Three Dimensions
The Sign of Approach to
Prosp ects for Future Work
A Some Geometrical Theorems
A Theorems and Pro ofs
A Application to SAWs
B Adequacy of Thermalization in the Pivot Algorithm
C Some Statistical Subtleties
D Remarks on the Field Theoretic Estimates of Universal Amplitude
Ratios
Introduction
The self avoiding walk SAW is a well known lattice mo del of a p olymer molecule
in a go o d solvent Its equivalence to the n limit of the n vector mo del
has also made it an imp ortant test case in the theory of critical phe
nomena
In this pap er we rep ort the results of an extensive Monte Carlo study of two
and three dimensional SAWs of length up to steps using the pivot algorithm
1
We make a high precision determination of the critical exp onents and
as well as several universal amplitude ratios In particular we make an
4
extremely sensitive test of the hyperscaling relation d which plays a
4
central role in the general theory of critical phenomena Section
Our results have also led us to reexamine critically the conventional theory
of p olymer molecules the so called two parameter renormalization group theory
Indeed such a reexamination is unavoidable as our Monte Carlo data are
inconsistent with this theory as it has b een heretofore applied But this is b ecause
as we explain in Section the theory has heretofore b een applied incorrectly
1
These computations were carried out over a year p erio d on a variety of RISC workstations
The total CPU time was several years but we have by now lost track of exactly how many
These p oints were rst made three years ago by Nickel in an imp ortant but
apparently under appreciated pap er they have recently b een extended by one of us
The Problem of Hyp erscaling
One of the key unsolved problems in the theory of critical phenomena has b een
the status of the so called hyperscaling relations scaling laws in which the spatial
dimension d app ears explicitly These relations have long b een known to rest on a
much more tenuous physical basis than the other scaling laws
Indeed it has b een understo o d since the early s that hyperscaling should not
hold for systems ab ove their upp er critical dimension d for d d the critical ex
u u
p onents are exp ected to b e those of mean eld theory and these exp onents satisfy
2
the hyperscaling relations only at d d For mo dels in an n vector univer
u
sality class including the SAW d equals It has generally b een b elieved that
u
hyperscaling should hold in dimensions d d but in our opinion there is no partic
u
ularly comp elling justi cation for such a b elief although the claim itself is probably
correct Hyp erscaling in the three dimensional Ising mo del is the sub ject of a con
3
troversy that has b een raging for years and which is still not completely settled
We remark that hyperscaling is also of interest in quantum eld theory where it
4
is equivalent to the nontriviality of the continuum limit for a strongly coupled
eld theory
Although the hyperscaling relations app ear naively to b e ineluctable conse
quences of the renormalization group approach to critical phenomena closer exami
nation reveals a mechanism by which hyperscaling can fail the so called dangerous
4
irrelevant variables But the much more di cult question of whether this viola
tion actually occurs in a given mo del can b e resolved only by detailed calculation
Unfortunately a direct analytical test of hyperscaling app ears to b e p ossible only
at or in the immediate neighborho o d of a Gaussian xed p oint that is for asymp
totically free theories or for small d d or large n
u
We note that the real space RG and eld theoretic RG frameworks
as typically used in approximate calculations implicitly assume the hyperscaling
relations so they cannot b e used to test hyperscaling
It is therefore of interest to make an unbiased numerical test of hyperscaling
working directly from rst principles One approach is series extrap olation
which a ords a direct test of universality and scaling laws including hyperscaling
2
This b elief has now b een con rmed by rigorous pro ofs of the failure of hyperscaling for the Ising
mo del in dimension d the self avoiding walk in dimension d and spread out
p ercolation in dimension d
3
See for series extrap olation work and
for Monte Carlo work
4
This mechanism was prop osed indep endently by Fisher and Wegner and Riedel p
and fo otnote in the early s For further discussion see also Ma Section VI I Amit and
Peliti Fisher App endix D and van Enter Fern andezand Sokal Section
It gives numerical results of apparently very high accuracy the claimed sub jec
tive error bars on critical exp onents are on the order of which
is comparable to the b est alternative calculational schemes However as is inherent
in any extrap olation metho d the results obtained dep end critically on the assump
tions made ab out the singularity structure of the exact function notably the nature
of the con uent singularity if any Indeed estimates by di er
ent metho ds from the same series sometimes di er among themselves by several
times their claimed error bars This together with systematic di erences b etween
lattices of the same dimension accounts for much of the controversy over hyper
scaling Quite a few extra terms would b e needed to resolve these discrepancies
in a convincing manner Unfortunately the computer time required to eval
uate the series co e cients grows exp onentially with the number of terms desired
while the extrap olation error is prop ortional to some inverse p ower of the number
of terms the p ower dep ends on the details of the correction to scaling terms and
the extrap olation metho d
In a Monte Carlo study by contrast one aims to prob e directly the regime
where the correlation length is For SAWs this corresp onds to a chain
length N The metho d a ords a direct test of universality and scaling laws
including hyperscaling In practice however it has b een extremely di cult to
obtain go o d data in the neighborho o d of the critical p oint There are two essential
di culties nite system size and critical slowing down For
spin mo dels and lattice eld theories these two factors together imply that the CPU
d+z d z
time needed to obtain one e ectively indep endent sample grows as L
where d is the spatial dimension of the system and z is the dynamic critical exp onent
5
of the Monte Carlo algorithm This situation may b e alleviated somewhat by
a new nite size scaling technique that yields accurate estimates of in nite
volume quantities from Monte Carlo data on lattice sizes L The situation
for SAWs is rather more favorable one can simulate a SAW directly in in nite
d
space with no nite size corrections or L factor in the CPU time There is to b e
sure critical slowing down but vast progress has b een made over the last decade
or so in inventing new and more e cient algorithms for simulating the SAW
In particular using the pivot algorithm one can generate an e ectively
indep endent N step SAW at least as regards global observables in a CPU time
of order N This is the b est p ossible order of magnitude since it takes
a time of order N merely to write down an N step walk Since N this
1
corresp onds to a CPU time in a spin system which if d is b etter
d+z
than even if z So the SAW is a uniquely favorable lab oratory for
studying the problem of hyperscaling
5
Conventional lo cal algorithms have z while the new collective mode algorithms
can have z and in some cases even z
Which Quantities are Universal
Over the past four decades various mathematical mo dels have b een employed
6
to describ e the b ehavior of linear p olymer molecules in a go o d solvent Among
these mo dels are the self avoiding walk the b ead ro d mo del and the con
tinuum Edwards mo del The detailed b ehavior dep ends on
the sp eci c mo del chosen just as the detailed b ehavior of real p olymer molecules
dep ends on the particular chemical structure of the p olymer and solvent and on the
temp erature However it has long b een understo o d that some asp ects of p olymer
b ehavior b ecome universal in the long chain limit N where N is the number
of monomers in the chain Unfortunately there has b een considerable confusion
ab out which quantities are universal and which are not In this subsection we sum
marize recent work of Nickel and one of us which clari es this issue
For further discussion see Section b elow
Standard renormalization group RG arguments predict that the mean
2 2
i and the i the mean square radius of gyration hR square end to end distance hR
g e
(mol)
2 2
N A of any real or mo del second virial co e cient A N M
Av og adr o 2
2
monomer
p olymer chain should have the asymptotic b ehavior
(1)
2 2
1
i A hR N b N
R
e
e R
e
(1)
2 2
1
hR i A N b N
R
g
g R
g
(mol) (1)
d
1
A A N b N
A
2
A
7
as N where d is the spatial dimension The critical exp onents and
1
(1) (1) (1)
are universal The amplitudes A b are nonuniversal in fact A b A b
R R A
e g
A R R
g e
(1) (1) (1)
even the signs of the correction to scaling amplitudes b b b and their various
R R A
e g
(1) (1) (1)
d b b combinations such as b are nonuniversal However the RG
R A
g
d 2
theory also predicts that the dimensionless amplitude ratios A A A A
R R A
g e
R
e
(1) (1) (1) (1)
are universal and b b b b
A R R R
e e g
So there is no reason why the correction to scaling amplitudes should have any
particular sign In the continuum Edwards mo del the e ective exp onents
e R
e
1 1
2 2
i d log N and i d log N and the interpenetration ratio d loghR d log hR
e R
g
e g
2 2
(mol)
d 2 2 d 2
d A hR i all approach their asymptotic values from below
2
g
(1) (1) (1)
that is b b and b On the other hand high
R R
e g
precision Monte Carlo data on lattice self avoiding walks see Section b elow
as well as Ref show clearly that these quantities approach their asymptotic
values from above and the same o ccurs in the b ead ro d mo del with su ciently large
b ead diameter Indeed this b ehavior is almost obvious qualitatively short self
avoiding walks b ehave roughly like hard spheres only at larger N do es one see the
6
Here go o d solvent means that we work at any xed temp erature strictly ab ove the theta
temp erature for the given p olymer solvent pair
7
In we have assumed for simplicity that the hyperscaling relation d is valid
4
softer excluded volume smaller characteristic of a fractal ob ject In any case
all these mo dels are in excellent agreement for the leading universal quantities
d 2
d 2
A A and d A A and they are in rough agreement for the
R R A
g e
R
g
(1) (1) (1) (1)
and b b b universal correction to scaling quantities b
1
A R R R
e e g
It is thus misguided to analyze the exp erimental data in the go o d solvent regime
by attempting to match the real p olymer molecules to the continuum Edwards
1 2
mo del via the corresp ondence z aN where a is an empirically deter
E dw ar ds
8
mined scale factor dep ending on the p olymer solvent and temp erature the con
tinuum Edwards mo del can predict only the universal quantities Indeed there is
evidence that real p olymers in a su ciently go o d solvent b ehave like
self avoiding walks i e they approach from ab ove in this case they cannot b e
matched to any value of z This b ehavior has heretofore b een considered
E dw ar ds
paradoxical in fact it is quite natural Hub er and Sto ckmayer attributed this
b ehavior to the e ects of chain sti ness In fact as p ointed out by Nickel chain
sti ness is quite irrelevant here as the e ect o ccurs also for perfectly exible chains
such as self avoiding walks or the b ead ro d mo del
These p oints have b een made previously by Nickel Similar comments have
b een made with regard to liquid gas critical p oints by Liu and Fisher
In summary the error of all two parameter theories is to fail to distinguish
correctly which quantities are universal and which are non universal In particular
the mo dern two parameter theory b egins from one sp ecial mo del the continuum
Edwards mo del and assumes incorrectly that it can describ e certain asp ects of
p olymer b ehavior e g the sign of approach to which in reality are non universal
Remark A very di erent limiting b ehavior is obtained in dimension d
if we take simultaneously N and T T such that x N T T remains
xed for a suitable crossover exp onent In a separate work one of us has
argued that it is precisely this universal crossover scaling behavior in an in nitesimal
region just ab ove the theta temp erature that is describ ed by the continuum Edwards
mo del
Plan of this Paper
The plan of this pap er is as follows In Section we review the needed back
ground information ab out the self avoiding walk and the pivot algorithm In Section
we analyze several algorithms for computing the second virial co e cient this sec
tion can b e skipp ed by readers whose main interest is in the results In Section
we present and analyze our Monte Carlo data for self avoiding walks in two and
three dimensions In Section we compare our results with previous work discuss
further the interpretation of the sign of approach to and discuss prosp ects for
d
the future In App endix A we prove some geometric b ounds for subsets of Z as a
corollary we prove hyperscaling for SAWs in dimension d In App endix B we
discuss the problem of ensuring adequate thermalization for the pivot algorithm
8
See for example the comparisons b etween theory and exp eriment in Section F and
Section
In App endix C we discuss some subtleties involved in the statistical analysis of our
data In App endix D we make a few remarks on the eld theoretic calculations of
universal amplitude ratios
Background and Notation
The Self Avoiding Walk SAW A Review
In this section we review brie y the basic facts and conjectures ab out the SAW
that will b e used in the remainder of the pap er Let L b e some regular d dimensional
lattice Then an N step self avoiding walk SAW on L is a sequence of distinct
p oints in L such that each p oint is a nearest neighbor of its predecessor
0 1 N
d
For simplicity we shall restrict attention to the simple hyper cubic lattice Z
similar ideas apply with minor alterations to other regular lattices We assume all
walks to b egin at the origin unless stated otherwise and we let S b e the
0 N
set of all N step SAWs starting at the origin and ending anywhere
First we de ne the quantities relating to the number or entropy of SAWs
d
Let c resp c x b e the number of N step SAWs on Z starting at the origin
N N
and ending anywhere resp ending at x Then c and c x are b elieved to have
N N
the asymptotic b ehavior
N 1
c N
N
N 2
sing
c x N x xed
N
as N here is called the connective constant of the lattice and and
sing
are critical exponents The critical exp onents are b elieved to b e universal among
lattices of a given dimension d For rigorous results concerning the asymptotic
b ehavior of c and c x see
N N
9
Next we de ne several measures of the size of an N step SAW
The squared end to end distance
2 2
R
N e
The squared radius of gyration
2
N N
X X
2
A
R a
j i
g
N N
i=0 j =0
2
N N
X X
2
b
i
i
N N
i=0 i=0
N
X
2
c
i j
2
N
i j =0
9
Some other measures of the size of a SAW will b e de ned in App endix A
The mean square distance of a monomer from the endpoints
N
h i
X
2 2 2
R
i N
m i
N
i=0
2 2 2
We then consider the mean values hR i hR i and hR i in the probability dis
N N N
e g m
tribution which gives equal weight to each N step SAW Very little has b een proven
rigorously ab out these mean values but they are b elieved to have the asymptotic
b ehavior
2 2 2 2
hR i hR i hR i N
N N N
e g m
as N where is another universal critical exp onent Moreover the ampli
tude ratios
2
i hR
N
g
A
N
2
i hR
N
e
2
i hR
N
m
B
N
2
i hR
N
e
10
are exp ected to approach universal values in the limit N Indeed the full
probability distribution of is exp ected to scale as
N
c x
N
d
N f x N
c
N
as N for a suitable scaling function f also universal mo dulo a single nonuni
11
versal scale factor and f is exp ected to b e rotation invariant All these b eliefs
can b e subsumed in the even more general assertion that the probability distribu
tion of the SAW with lengths rescaled by N converges weakly as N to some
12
well de ned probability measure on a space of continuum chains
(1) (2) (1)
b e the number of pairs such that is an N step Finally let c
1 N N
1 2
(2) (1)
SAW starting at the origin is an N step SAW starting anywhere and and
2
(2) (1) (2)
have at least one p oint in common i e Equivalently we can
in terms of walks that b oth start at the origin write c
N N
1 2
X X
(1) (2)
T c
N N
1 2
(1) (2)
S S
N N
1 2
10 2 2
Sometimes the notation hR i hR i is used instead
g e
11
Actually is claimed to hold only for jxj of order N The precise statement of is
therefore that the limit
c N y
N
d
f y lim N
N !1
c
N
exists for each y and that f y
12
Very recently Hara and Slade have proven that the SAW in dimension d converges
1 2
weakly to Brownian motion when N with lengths rescaled by CN for a suitable nonuniver
1
and also that have the sal constant C It follows from this that holds with
2
1 1
limiting values A B Earlier Slade had proven these results for su ciently
1 1
6 2
high dimension d See also
where
(1) (2) d (1) (2)
T fx Z x g
(2) (1)
is the number of translates of that somewhere intersect It is b elieved that
N +N (2 + 2) 2
1 2 4
c N N g N N
N N 1 2 1 2
1 2
as N N where is yet another universal critical exp onent and g is a
1 2 4
universal scaling function
The quantity c is closely related to the second virial co e cient To see this
N N
1 2
consider a rather general theory in which molecules of various types interact Let
the molecules of type i have a set S of internal states so that the complete state
i
d
of such a molecule is given by a pair x s where x Z is its p osition and s S is
i
its internal state Let us assign Boltzmann weights or fugacities W s s S
i i
P
W s and let us assign an to the internal states normalized so that
i
s S
i
d
x s x s x x Z s S s S to a molecule of interaction energy V
i j ij
type i at x s interacting with one of type j at x s Then the second virial
co e cient b etween a molecule of type i and one of type j is
i h
X X
0 0
(ij )
V ((0 s) (x s ))
ij
B W s W s e
i j
2
0 d
x Z
s S
i
0
s S
j
In the SAW case the types are the di erent lengths N the internal states are the
conformations S starting at the origin the Boltzmann weights are W
N N
c for each S and the interaction energies are hard core repulsions
N N
if x x
0
V x x
NN
otherwise
It follows immediately that
c
N N
(N N )
1 2
1 2
B
2
c c
N N
2 1
(N N )
1 2
The second virial co e cient B is a measure of the excluded volume b e
2
tween a pair of SAWs It is useful to de ne a dimensionless quantity by normalizing
(N N )
1 2
2
B by some measure of the size of these SAWs Theorists prefer hR i as the
2
e
2
measure of size while exp erimentalists prefer hR i since it can b e measured by light
g
scattering We thus de ne the theorists interpenetration ratio
(N N )
c B
N N
2
d 2 d 2
d d
R N
d 2 d 2
2
2 2
i i hR c hR
e e N N N
and the usual interpenetration ratio
(N N )
c B
N N
2
d 2 d 2
d d
N
d 2 d 2
2
2 2
i i hR c hR
N N g g N
for simplicity we consider only N N N The numerical prefactors are a
1 2
convention that arose historically for reasons not worth explaining here Crudely
sp eaking measures the degree of hardness of a SAW in its interactions with
other SAWs A useful standard of comparison is the hard sphere of radius r and
constant density
d 2
d
r B
2
d d
d
2 2
R r
g
d
and hence
in d
d 2
d
in d
har d spher e
in d d d
in d
Inserting and into we see that
2 d
4
N
N
as N We can therefore distinguish three a priori p ossibilities
a d so that as N This b ehavior cannot o ccur
4 N
unless typical SAWs have a very strange p orcupine like shap e which is quite
13
implausible
b d In the simplest case this means that
4 N
as N i e typical SAWs exclude each other within a constant factor
like hard spheres This b ehavior is called hyperscaling However the relation
d is also consistent with a logarithmic violation of hyperscaling
4
p
i e log N as N for some p ower p
N
c d so that as N This is a p ower law violation
4 N
of hyperscaling typical SAWs exclude each other in nitely more weakly than
hard spheres
A very b eautiful heuristic argument concerning hyperscaling for SAWs was given
by des Cloizeaux Note rst from that measures roughly sp eaking
the probability of intersection of two indep endent SAWs that start a distance of
2 1 2
order hR i N apart Now by we can interpret a long SAW as an ob ject
g
with fractal dimension Two indep endent such ob jects will generically
intersect if and only if the sum of their fractal dimensions is at least as large as the
13
Mo dulo some reasonable assumptions this b ehavior can in fact b e rigorously excluded see
Theorem A and equations A A in App endix A
dimension of the ambient space So we exp ect lim to b e nonzero if and
N
N
only if
d i e d
Since it is b elieved that
1
for d
2
1 8
1
log for d
2
2
1 15
for d
2 16 512
for d this pap er
3
for d
4
we see that
d for d
d logs for d
d for d
Therefore we exp ect
hyperscaling for d
logarithmic violation of hyperscaling for d
p ower law violation of hyperscaling for d
One half of this heuristic argument can b e proven rigorously It is easy to see
that
c N N c c
N 1 2 N N N
2 1 1 2
so that
(N N )
1 2
N N B
1 2
2
or in terms of critical exp onents
4
It follows that d implies d i e the p ower law violation of
4
hyperscaling This is now proven rigorously to o ccur for d
In the p olymer physics literature it is usually taken for granted that hyperscaling
holds in dimension d But in our opinion hyperscaling is a deep prop erty that
needs to be tested
We remark that dimension d is a di erent case here hyperscaling can b e
proven rigorously mo dulo some reasonable assumptions on the scaling of individual
SAWs We present this pro of in App endix A it is the analogue for SAWs of
Aizenman s pro of Section of hyperscaling for two dimensional Ising mo dels
with nite range ferromagnetic interaction The underlying geometric idea is that
SAWs in the plane cannot avoid intersecting each other
Finally we need to make some comments ab out corrections to scaling Clearly
are only the leading term in a large N asymptotic expan
sion According to renormalization group theory the mean value of any global
observable O b ehaves as N as
a b b b a
2 0 1 2 1
p
hO i AN
N
2 +1 +2
1 1 1
N N N N N
c c c
0 1 2
+1 +2
2 2 2
N N N
k
Thus in addition to analytic corrections to scaling of the form a N there are
k
+k +k
1 2
non analytic corrections to scaling of the form b N c N and so forth
k k
as well as more complicated terms not shown in which have the general form
k +k + +l
1 1 2 2
const N where k k and l are non negative integers The leading
1 2
exp onent p and the correction to scaling exp onents are universal p
1 2
of course dep ends on the observable in question but the do not Please note
i
that the exp onents have no relation whatso ever to the gap exp onent
1 2
de ned in The notation used here is standard but unfortunate The
4
various amplitudes b oth leading and subleading are all nonuniversal However
ratios of the corresp onding amplitudes A b and c but not a or the higher b c
0 0 k k k
for di erent observables are universal
Remark The names of the critical exp onents and are chosen by
sing 4
analogy with the corresp onding exp onents in ferromagnetic spin systems
Indeed the generating functions of self avoiding walks
X
N
c
N
N =0
X
N
G x c x
N
N =0
X
N +N
1 2
c u
N N 4
1 2
N N =0
1 2
are equal to the susceptibility spin spin correlation function and fourth cumulant
in the n vector mo del analytically continued to n In particular if x
is a nearest neighbor of the origin then G x is essentially the energy E up to
an additive and multiplicative constant The quantity
P
1 2
2
jxj G x
B C
x
P
A
G x
x
is the second moment correlation length Inserting into
we obtain the leading b ehavior
c
1
sing
G x regular terms
c
c
2
4
u
4 c
as approaches the critical point Note in particular that is the
c sing
exp onent for the singular part of the sp eci c heat C E the exp onent for
H
the full sp eci c heat is max If the hyperscaling relation d
sing 4
holds without multiplicative logarithmic corrections then the renormalized cou
2 d
pling constant g u tends to a nonzero limiting value g as so
4 c
hyperscaling without multiplicative logarithmic corrections can b e interpreted as
the non Gaussianness nontriviality of the scaling limit quantum eld theory
p
The Pivot Algorithm A Review
The pivot algorithm was invented in by Lal reinvented in by
MacDonald et al and again reinvented a short time later by Madras The
pivot algorithm is the most e cient algorithm currently known for simulating SAWs
in the xed N variable x ensemble Here we summarize brie y the relevant features
of the algorithm more details can b e found in
The elementary move of the pivot algorithm is as follows Cho ose at random a
pivot p oint k along the walk k N choose at random an element g of the
symmetry group of the lattice rotation or re ection or a combination thereof then
apply g to the part of the walk subsequent to the pivot p oint namely
k +1 N
using as the temp orary origin That is the prop osed new walk is
k
for i k
i
i
g for k i N
k i k
The walk is accepted if it is self avoiding otherwise it is rejected and the old walk
is counted once more in the sample It is easy to see that this algorithm satis es
detailed balance for the standard equal weight SAW distribution Ergo dicity is less
obvious but it can b e proven
At rst thought this seems to b e a terrible algorithm for N large nearly all the
prop osed moves will get rejected In fact this latter statement is true but the hasty
conclusion drawn from it is radically false The acceptance fraction f do es indeed
p
go to zero as N roughly like N empirically it is found that the exp onent p
is in d and in d But this means that roughly
p
once every N moves one gets an acceptance And the pivot moves are very radical
one might surmise that after very few accepted moves say or the SAW will
have reached an essentially new con guration One conjectures therefore that
p
the auto correlation time of the pivot algorithm b ehaves as N Things are
in fact somewhat more subtle see the next paragraph but roughly sp eaking and
mo dulo a p ossible logarithm this conjecture app ears to b e true On the other hand
a careful analysis of the computational complexity of the pivot algorithm see also
b elow shows that one accepted move can b e pro duced in a computer time of order
N Combining these two facts we conclude that one e ectively indep endent
sample can b e pro duced in a computer time of order N or p erhaps N log N
Let s lo ok more closely Supp ose we know that the acceptance fraction f in the
p
pivot algorithm b ehaves as f N as N Then as argued ab ove after a
p
few successful pivots i e a time of order f N the global conformation of
the walk should have reached an essentially new state Thus we exp ect that for
2 2
observables A which measure the global prop erties of the walk such as R R or
e g
2
R the auto correlation time see C should b e a few times f This
int A
m
is con rmed numerically Section On the other hand it is imp ortant to
th th
recognize that local observables such as the angle b etween the and steps
of the walk may evolve a factor of N more slowly than global observables For
example the observable mentioned in the preceding sentence changes only when
17
serves as a successful pivot p oint and this happ ens on average only once every N f
attempted moves Thus for local observables A we exp ect to b e of order N f
int A
General prop erties of reversible Markov chains then imply that the exp onential
auto correlation time must b e of at least this order and if we have not overlooked
exp
any slow mo des in the system then should b e of exactly this order Finally
exp
even the global observables are unlikely to b e precisely orthogonal to the slowest
mo de so it is reasonable to exp ect that b e of order N f for these observables
exp A
to o In other words for global observables A we exp ect the auto correlation function
t to have an extremely slowly decaying tail which however contributes little
AA
to the area under the curve This b ehavior is illustrated by the exact solution of
the pivot dynamics for the case of ordinary random walk Section and by
numerical calculations for the SAW see App endix C
Computational complexity A very imp ortant issue in any algorithm but
esp ecially in a non lo cal one is the CPU time p er iteration By using a hash
table the self avoidance of a prop osed new walk can b e checked in a time
of order N But one can do even b etter by starting the checking at the pivot p oint
1 p
and working outwards failures can b e detected in a mean time of order N
1 p
Sections and The mean CPU time per successful pivot is therefore N
p
for each of N failures plus N for one success or N in all Combining this with
the observations made previously we conclude that one e ectively indep endent
sample as regards global observables can b e pro duced in a computer time of
order N
Initialization There are two main approaches
Equilibrium start Generate the initial con guration by dimerization
then the Markov chain is in equilibrium from the b eginning and no data need b e
discarded This approach is feasible and recommended at least up to N of order
a few thousand There is no harm in sp ending even days of CPU time on this
step provided that this time is small compared to the rest of the run after all the
algorithm need only b e initialized once
Thermalization Start in an arbitrary initial con guration and then dis
card the rst n N f iterations This is painful b ecause is a factor
disc exp exp
5
N larger than for global observables A thus for very large N the
int A
CPU time of the algorithm could end up b eing dominated by the thermalization
Nevertheless one must resist the temptation to cut corners here as even a small
initialization bias can lead to systematically erroneous results esp ecially if the sta
tistical error is small see App endix B for striking evidence of this Some mo dest
gain can probably b e obtained by using closer to equilibrium initial con gurations
e g but it is still prudent to take n at least several times N f
disc
Initialization will b ecome a more imp ortant issue in the future as faster com
puters p ermit simulations at ever larger chain lengths
Algorithms for Counting Overlaps
In this section we discuss algorithms for computing the excluded volume b etween
a given pair of SAWs this is the key step in a Monte Carlo study of the second
virial co e cient This section can b e skipp ed by readers whose main interest is in
the results rather than the algorithms
Generalities
(1) (2) (1) (2)
Let and b e resp ectively N step and N step SAWs and de ne T
1 2
(2) (1)
to b e the number of translates of which somewhere intersect
(1) (2) d (1) (2)
T fx Z x g a
(1) (2)
b
(1) (2)
where A B fy z y A z B g The exp ected value of T averaging
(2) (1)
This quantity c c is c and S over indep endent walks S
N N N N N N
2 1 1 2 2 1
has the asymptotic b ehavior
(2 ) 2
4
N N g N N c c c
1 2 1 2 N N N N
2 1 1 2
where g is a scaling function cf It is thus p ossible to estimate the
critical exp onent by running two indep endent pivot algorithms and mea
4
(1) (2)
suring T Typically one would run at N N N for a sequence of
1 2
values of N In particular this allows a direct Monte Carlo test of the hyperscaling
relation d Note that an indep endent measurement of is not needed
4
The e cient determination of T A A A A for a sp eci ed pair of
1 2 1 2
d
sets A A Z is a very interesting and nontrivial problem in computer science
1 2
We see two broad approaches
Deterministic algorithms which compute T A A exactly
1 2
Monte Carlo algorithms which pro duce an unbiased or almost unbiased es
timate of T A A
1 2
In the latter case the statistical uctuations in the auxiliary inner lo op Monte
Carlo pro cess would b e added to those in the main Monte Carlo program but this
is acceptable provided that the former are not to o large compared to the latter see
Section
We shall discuss the deterministic algorithms for computing T A A in Section
1 2
and the Monte Carlo algorithms in Sections
Notation
We shall denote by N resp N the number of p oints in the set A resp
1 2 1
A We also write N min N N and N max N N An N step
2 min 1 2 max 1 2
self avoiding walk has N N p oints
d
Now x a pair of sets A A Z and write
1 2
S A A fy z y A z A g
1 2 1 2
Our goal is to compute T A A S An imp ortant role is played by the
1 2
function
x f y z y A z A y z xg a
1 2
x b
A A
2 1
where and are the indicator functions of the sets A and A resp ectively
A A 1 2
1 2
and z z and denotes convolution Clearly
A A
2 2
x for x S a
x N for x S b
min
We also write
if x if x S
I x
if x if x S
Note that
X X
T A A I x
1 2
d
x S
x Z
X X
x x N N
1 2
d
x S
x Z
For future reference we de ne also
X
U A A
1 2
x
x S
This observable has little intrinsic interest but it will play an imp ortant role in
14
two of the Monte Carlo algorithms It is not hard to see that
N N T A A N N
1 2 1 2 1 2
2
T A A
1 2
U A A T A A
1 2 1 2
N N
1 2
Moreover we shall prove in App endix A Theorem A that
U A A log N N
1 2 1 2
We shall use h i to denote exp ectation with resp ect to some probability distri
bution on pairs of sets A A e g the equal weight ensemble on the space S S
1 2 N N
1 2
of pairs of SAWs We shall use E to denote exp ectation with resp ect to some
inner lo op Monte Carlo algorithm
Deterministic Algorithms
Here we introduce some deterministic algorithms for computing x I x
T A A and or U A A These algorithms can b e employed either stand
1 2 1 2
alone or as building blo cks for the Monte Carlo algorithms to b e introduced later
Supp ose rst that we want to compute x for a single value of x This
can b e done as follows write all the p oints of A into a hash table then
1
examine sequentially each of the p oints z A inquiring whether y x z b elongs
2
to A and incrementing a counter if it do es Clearly this requires a CPU time of
1
order N N
1 2
For computing I x the algorithm can b e streamlined by stopping as so on as
one nds x
Supp ose next that we want to compute x for several di erent values of x
say r of them Then the foregoing algorithm requires a CPU time of order N r N
1 2
But since the answer is invariant under the interchange A A x x it now
1 2
pays to choose A to b e the smal ler of the two sets As a result the CPU time is
2
of order N r N
max min
Supp ose nally that we want to compute x or I x for al l x A sp ecial
case of this is to compute T A A or U A A This can b e done by examining
1 2 1 2
sequentially each of the pairs y A z A and writing the p oints x y z into
1 2
a hash table equipp ed with an auxiliary count eld Clearly this requires a CPU
time of order N N The CPU time for subsequently utilizing the counts f x g is
1 2
14
The two upp er b ounds are trivial The lower b ound in is the Schwarz inequality The
lower b ound in is proven as follows Let y resp z b e the lexicographically smallest element
of A resp A Then y A y z A z so y A and A z are subsets of S of
1 2 2 1 2 1
cardinalities N and N resp ectively with only one p oint in common namely y z Q E D
2 1
of order T A A N N If all one wants is fI x g then the count eld can b e
1 2 1 2
disp ensed with it su ces to know which sites x are hit at least once
An alternative algorithm for computing x for all x can b e based on the
th
Fast Fourier Transform FFT Let j d b e the extension of A in the j
j 1
co ordinate direction i e the di erence b etween the maximum and minimum values
of y for y A Let b e the corresp onding extension for A Now let k b e the
j 1 j 2 j
k
j
least integer such that Then if we place A and A in a p erio dic
j j 1 2
k k k
1 2
d
b ox B of size we can compute the convolution of and
A A
1 2
in the b ox B without distortion by the p erio dic b oundary conditions this gives
the full set of counts f x g and thus also fI x g T A A and U A A The
1 2 1 2
convolution can b e carried out by the FFT in a time V log V where
d
Y
k +...+k
1
d
V
j j
j =1
Let us now estimate the p erformance of these metho ds as stand alone algo
(1) (2) (1) (2)
rithms for computing T where and are self avoiding walks Al
(1) (2)
gorithm computes T in a CPU time of order N N N N
1 2 1 2
2
i e of order N if N N N By contrast we exp ect that one e ectively inde
1 2
(1) (2)
p endent sample of the pair can b e pro duced by the pivot algorithm in
p
a CPU time of order N N if N or in any case not much greater So
1 2 int T
this algorithm would sp end more time analyzing the data than pro ducing it and
the overall computational complexity p er e ectively indep endent sample would
2
b e increased from N to N thereby nullifying the advantage of the pivot algorithm
over previous algorithms
(1) (2)
Algorithm computes T in a CPU time of order V log V where V
d
is given by For typical SAWs we have V N N so presumably
1 2
we have
d
hV log V i N N log N N
1 2 1 2
In the usual situation N N N this metho d is asymptotically b etter than
1 2
algorithm if d i e if d However it is unlikely to b e b etter in practice
except for very large N And the b ehavior is still vastly worse than the time of
order N for generating the walks except in d
It may b e p ossible to devise deterministic algorithms which are more e cient
than either of these elementary ones we leave this as an exercise for interested
computer scientists
Monte Carlo Algorithms Generalities
Fix a pair of sets A A and supp ose that we use some Monte Carlo algorithm
1 2
to provide an unbiased estimate Z of T A A We will thus have
1 2
E Z jA A T A A a
1 2 1 2
2 2
var Z jA A E Z jA A E Z jA A V A A b
1 2 1 2 1 2 1 2
Here a expresses the unbiasedness of the inner lo op Monte Carlo algorithm
while b de nes its conditional variance We shall compute the functional
V A A for each of the Monte Carlo algorithms we introduce Sections
1 2
Now let us call the Monte Carlo subroutine R times for the given pair A A
1 2
P
R
1
Z Obviously we have and average the results Z R
i
i=1
E Z jA A T A A a
1 2 1 2
1
var Z jA A R V A A b
1 2 1 2
Now supp ose that we generate a random pair A A from some probability
1 2
(1) (2)
distribution e g SAWs from the equal weight distribution on S S
N N
1 2
Clearly Z is an unbiased estimator of hT i i e
hZ i hE Z jA A i hT i
1 2
where h i denotes exp ectation in the given probability distribution The variance
of Z is a sum of two terms
var Z var T hvar Z jA A i
1 2
i h
1 2 2
R hV i hT i hT i
The rst term is the uctuation of T A A from one pair of sets to another the
1 2
second term is the mean over pairs of sets of the uctuation conditional variance
in the inner Monte Carlo subroutine
The mean CPU time for the computation of Z is hT i a bR here a is
CPU
the mean CPU time for generating a pair of e ectively indep endent sets A A
1 2
from the desired ensemble plus any setup time asso ciated with the inner lo op
Monte Carlo algorithm while b is the mean additional CPU time p er iteration of
the inner lo op Monte Carlo algorithm The goal is to minimize the variance time
pro duct
1
hT i var Z b var T R ahV iR a var T bhV i
CPU
since this quantity divided by the total CPU time equals the variance of our nal
estimate Hence the optimal choice of R is
1 2
a hV i
R
opt
b var T
and the variance time pro duct is then
i h
2
1 2 1 2
hT i var Z a var T b hV i
CPU opt
Of course R must b e a p ositive integer and so the true R is obtained by rounding
opt
the right hand side of up or down This subtlety can b e ignored if R is
opt
large but may b e signi cant otherwise In particular the deterministic inner lo op
algorithms Section have V but in this case R rather than as
opt
claims and hT i var Z a b var T
CPU opt
In the remainder of this section we will assume that the CPU time for generating
the sets A and A is of order N N Clearly this is a lower b ound since it takes a
1 2 1 2
time of order N N simply to write down the two sets On the other hand for our
1 2
application to SAWs A and A will b e generated by the pivot algorithm which
1 2
generates an e ectively indep endent SAW as regards global observables in a
(1) (2)
CPU time of order N see Section We exp ect that the observable T
p
is indeed global in the sense that N where p is the acceptance fraction
int T
exp onent
Hit or Miss Monte Carlo Algorithm
+
th
Let resp b e the maximum resp minimum value of the j co ordinate
j j
among the p oints in A so that
1
+ +
B
1
d d 1 1
+
is the smallest rectangular parallelopip ed containing A Let and b e the
1
j j
corresp onding values for A and B the corresp onding b ox It follows that
2 2
+ + + +
B B B
1 2
d d d d 1 1 1 1
15
is a parallelopip ed which is guaranteed to contain all the p oints of S A A
1 2
P
I x can b e computed by the trivial hit Therefore T A A S
1 2
x B
or miss Monte Carlo metho d Pick a p oint x B at random and compute I x by
the deterministic algorithm of Section then output Z B I x where
Q
+ d +
Clearly I x is and B Here
j j j j
j j j j
j =1
a binomial random variable of mean p S B so that
S
E I x
B
S S
var I x
B B
Hence Z B I x is an unbiased estimator of S and its variance is
V A A var Z S B S
hit or miss 1 2
The CPU time for R iterations of this algorithm is of order T N
CPU max
RN we put the larger of the two sets A A in the hash table once and then
min 1 2
each time we compute x by lo oping over the smaller of the two sets The CPU
time for generating the two sets is by assumption also of order N Therefore in
max
we have a N and b N
max min
15
In fact it is the smallest such parallelopip ed since for each index j there is a p oint in A A
1 2
+ +
th th
with j co ordinate equal to and another p oint with j co ordinate equal to
j j j j
Barrett Algorithm Theory
Barrett has prop osed the following Monte Carlo algorithm which gives an
unbiased estimate of T A A
1 2
Cho ose at random y A and z A Set x y z
1 2
Compute x using the deterministic algorithm describ ed in Section
Note that by construction we have x S and hence x
Output Y N N x
1 2
The analysis of this algorithm is easy In Step we choose the vector x with
probability
x
Prob x
N N
1 2
It follows that
X X
E Y Prob x Y x T A A a
1 2
x S x S
X X
N N
1 2
2 2
N N U A A b E Y Prob x Y x
1 2 1 2
x
x S x S
and hence
2
V A A var Y N N U A A T A A
B ar r ett 1 2 1 2 1 2 1 2
The CPU time for R iterations of the Barrett algorithm is of order T
CPU
N RN we put the larger of the two sets A A in the hash table once
max min 1 2
and then each time we compute x by lo oping over the smaller of the two sets
The CPU time for generating the two sets is by assumption also of order N
max
Therefore in we have a N and b N
max min
Karp Luby Algorithm Theory
Karp and Luby have devised an elegant Monte Carlo algorithm for
estimating T A A and somewhat more general combinatorial problems The
1 2
Karp Luby algorithm go es as follows
Cho ose at random y A and z A Set x y z Set t
1 2
Cho ose at random y A
1
If z y x A then go to Step Otherwise increment t by and go to
2
Step
Output Z N t
2
This algorithm can b e understo o d as a randomized version of the Barrett algorithm
Step is identical in the two algorithms and it selects the vector x with probability
x
Prob x
N N
1 2
Then t the number of trials of Steps and needed to nd a y such that y x A
2
is conditioned on x a random variable with a geometric distribution
k 1
x x
for k Prob t k jx
N N
1 1
Hence the conditional exp ectations of Z N t are
2
N N
1 2
a E Z jx
x
2
N N N
1 1
2
2
E Z jx b
x x
Of course this makes sense only for x S Thus Steps of the Karp Luby
algorithm pro duce a random quantity Z whose mean value conditional on x is
precisely the deterministic quantity Y N N x of the Barrett algorithm It
1 2
follows that the unconditional exp ectations are
X
E Z Prob x E Z jx T A A a
1 2
x S
X X
N
1
2 2
E Z Prob x E Z jx N
2
x
x S x S
N N U A A N T A A
1 2 1 2 2 1 2
b
and hence
2
V A A var Z N N U A A N T A A T A A
K ar p Luby 1 2 1 2 1 2 2 1 2 1 2
For future reference we note the following inequality
N
1
N N U A A V A A N N U A A
1 2 1 2 K ar p Luby 1 2 1 2 1 2
N N
1 2
Proof The upp er b ound is trivial To prove the lower b ound use and
2
to deduce N N T T N N U This means that V is
1 2 1 2 K ar p Luby
of the same order of magnitude as its rst term namely N N U except p erhaps
1 2
when N N An alternative and often sharp er lower b ound on V can
1 2 K ar p Luby
b e obtained by noting that
1 2
N
1
log N N N U N T
max 1 2 2
N
2
2
Proof From we have T N N U while from we have log N
1 2 max
U Now take the geometric mean of these two b ounds Therefore we have
1 2
N
1
2
log N N N U A A T A A V A A
max 1 2 1 2 1 2 K ar p Luby 1 2
N
2
The CPU time for one execution of Steps and is essentially t so the exp ected
CPU time for one iteration of the Karp Luby algorithm is E t T A A N In
1 2 2
addition there is an initial CPU time of order N to place the elements of A in a
2 2
hash table Finally we should remember the time of order N N for generating A
1 2 1
and A in the rst place The exp ected CPU time for R iterations of the Karp Luby
2
algorithm plus generating A and A is thus
1 2
T A A
1 2
T N R
CPU max
N
2
Therefore in we have a N and b hT i N
max 2
Scaling Theory
In this section we consider the scaling theory of the three Monte Carlo algorithms
hit or miss Barrett and Karp Luby in the case where A and A are indep en
1 2
dent random SAWs of lengths N and N resp ectively and N N N In
1 2 1 2
each case we need to compute or guess heuristically the scaling b ehavior of var Z
and hT i as N A go o d gure of de merit for an algorithm is the mean
CPU
CPU time needed to estimate hT i with a relative variance of order this time is
hT i var Z
CPU opt
2
hT i
2
For the hit or miss algorithm we must study the scaling of hT i hT i and hT B i
2
For the Barrett and Karp Luby algorithms we must study the scaling of hT i hT i
and hU i as well as their various combinations
As discussed in Section we exp ect that hT i scales as
p
T
hT i N
where
p
T 4
We further exp ect that the probability distribution of T will after rescaling by
p
T
approach a nontrivial limiting distribution here nontrivial means that the N
distribution is not a delta function Therefore we exp ect that
2 2p
T
hT i var T N
On the other hand it is reasonable to exp ect that
p
U
hU i N
for some a priori unknown exp onent p From we know that p
U T
p p and we will b e interested in knowing whether these inequalities are strict
U T
or not
We now lo ok at the individual algorithms
Hit or Miss Algorithm As just discussed we exp ect
p
T
hT i N
2p
T
var T N
with p On the other hand for typical pairs of SAWs we clearly have
T 4
d
B N so we exp ect
d +p
T
hT B i N
In general we have d p but even when d p i e hyperscaling holds it
T T
2
seems intuitively clear that the ratio hT i hT B i will stay well b elow except
in dimension d Therefore radical cancellations in are excluded and we
exp ect
d +p
T
hV i N
hit or miss
It follows from and that
2p 1 d +p
T T
var Z N R N
On the other hand the CPU time is obviously
hT i N RN
CPU
The variance time pro duct is thus
i h
2p d +p 1 2p d +p
T T T T
hT i var Z N N R N R N N
CPU
There are two cases
a If the hyperscaling relation d p holds then R We exp ect that
T opt
2
this is the case for d With this choice of R we have var Z hT i and
hT i N In other words we can achieve a relative variance of order in a CPU
CPU
time of order N
(d p ) 2
T
b If d p then R N We exp ect that this is the case for
T opt
d More precisely for d we exp ect that and p in which
T
(d 4) 4 3 2 d 2
case R N With this choice of R we have var Z hT i N and
opt
1+d 2 1 2 1+d
hT i N hT i which implies a variance time pro duct of order N hT i
CPU
In fact a variance time pro duct of this order can b e achieved with any R in the
d p
T
range R N the optimal value R lies at the geometric mean of these
opt
two extremes Hence a relative variance of order requires a CPU time of order
1+d p 1+(d 4) 2
T
N i e order N for d
Barrett and Karp Luby algorithms The b ehavior of the Barrett and Karp
2
Luby algorithms is determined by the scaling b ehavior of hT i hT i and hU i as
N From and we have
2
hT i
hV i N N hU i
B ar r ett 1 2
N N hU i
1 2
2
hT i
hV i N N hU i
K ar p Luby 1 2
N N hU i
1 2
We write a b to denote that a b as N This fo cusses attention on the
ratio
2
hT i
N N
1 2
N N
1 2
N N hU i
1 2 N N
1 2
which by satis es N N and in particular on
1 2
lim N N
N
For simplicity we consider here only N N N but the same principles ob
1 2
viously apply at any xed ratio N N There are then three p ossible
1 2
cases
2 2
a In this case hV i N hU i and hV i N hU i
B ar r ett K ar p Luby
2
b In this case hV i N hU i and hV i
B ar r ett K ar p Luby
2
N hU i
2
c In this case hV i N hU i and its exact scaling is subtle Of
B ar r ett
2
course we still have hV i N hU i
K ar p Luby
Case a corresp onds to the exp onent p b eing strictly greater than p while
U T
cases b and c corresp ond to p p We b elieve that in fact case c never
U T
2 (1) (2)
o ccurs although it is p ossible to have U T N N for sp ecial pairs e g
1 2
p erp endicular ro ds it seems quite implausible that such b ehavior could o ccur even
in the limit N after averaging over all pairs of SAWs
In Section we shall present numerical data showing clearly that in
dimensions d what is less clear is whether is zero or nonzero For dimension
2 16
d we exp ect that For d we exp ect that hT i N from which
2 17
it follows that hU i N and hence On the other hand we b elieve as
noted ab ove that is imp ossible
Assuming that it follows that hV i and hV i are b oth of
B ar r ett K ar p Luby
2
order N hU i Therefore
2 1 2
var Z hT i R N hU i
16
This is rigorously proven for d mo dulo the problem of translating generating function
results into xed N results See Theorem and Remark following it
17 2 2 2 2
Proof hU i hT i N hT i N by and the Schwarz inequality while
2 2
hU i hT i by Thus if hT i N we have also hU i N
for b oth algorithms On the other hand the CPU times are
hT i N NR a
C P U B ar r ett
1
hT i N N hT iR b
C P U K ar p Luby
The variance time pro ducts are thus
h i
2 2 1 2 2
hT i var Z N hT i R N hU iR hT i N hU i
C P U B ar r ett
a
h i
2 3 2 1 2
hT i var Z N N hT i R N hU iR hT i hU ihT i
C P U K ar p Luby
b
2 2
Recalling now the inequality T N U cf we see that the dom
inant term for the Barrett algorithm is the fourth one provided that R
2 2 3
N hU i hT i and in this case we have hT i var Z N hU i The optimal value
CPU
1 2
is at the geometric mean of this range i e R N hU i hT i We conclude that
opt
3 2
a relative variance of order requires a CPU time of order N hU i hT i This is
2 2
at least of order N but it may b e larger in case hU i hT i N i e in case
p p
U T
Recalling next the inequality U T cf we see that the dominant term
2 2
R for the Karp Luby algorithm is the third one provided that N hU i hT i
2 2
N hT i and in this case we have hT i var Z N hT i The optimal value is at
CPU
2 1 2 3 2
the geometric mean of this range i e R N hU i hT i We conclude that a
opt
relative variance of order requires a CPU time of order N
Comparing the analyses of the Barrett and Karp Luby algorithms we see that
the e ect of the additional randomization in the Karp Luby algorithm is to reduce
drastically the mean CPU time p er iteration T N versus N while only mo d
2 min
estly increasing the variance The Karp Luby algorithm is thus sup erior to the
2
Barrett algorithm whenever hT i N as we exp ect o ccurs in dimension d
2
the two algorithms are of the same order whenever hU i hT i N as we exp ect
o ccurs in dimension d
Barrett and Karp Luby Algorithms Numerical Results
In this section we rep ort results of a rather crude Monte Carlo study of hT i
2
hT i var T hU i hV i and hV i for SAWs in dimensions d and
B ar r ett K ar p Luby
d taking N N N The goal is to estimate the exp onents p and p
1 2 T U
and the various amplitudes in the scaling relations
p
T
hT i A N a
T
2p 2
T
2
N b hT i A
T
2p
T
var T A N c
v ar (T )
p
U
hU i A N d
U
2+p
U
hV i A N e
B ar r ett V
B ar r ett
2+p
U
hV i A N f
K ar p Luby V
K ar p Luby
In Table we rep ort our Monte Carlo estimates for dimensional self avoiding
walks at N These were obtained by using the dimer
(1) (2)
ization algorithm to generate pairs of indep endent SAWs and then using
(1) (2) (1) (2)
deterministic algorithm Section to compute T and U
Unfortunately some of the observables lack error bars this is our fault The t
2
to hT i gives a reasonable provided that we discard the data p oint at the lowest
value of N we then get p This is not far from the exp ected
T
value p d the small discrepancy can quite plausibly b e attributed to
T
corrections to scaling We will make a much more careful estimate of hT i and its
2
exp onent p in Section The ts to hT i and var T are also consistent with
T
p although the lack of error bars prevents making this quantitative The
T
2
t to hV i gives a reasonable only if we discard the data p oints at the
K ar p Luby
two lowest values of N we then get p If we use the same data
U
p oints in the ts to hU i and hV i we get p and unfortunately
B ar r ett U
without error bars These estimates of p are thus in rather medio cre agreement
U
and it is far from clear whether p is strictly greater than p Another way
U T
2 2
of lo oking at this is to study the ratio hT i N hU i which decreases from
N
at N to at N it is far from clear whether this ratio
is tending to zero or to a nonzero value as N Clearly data at much larger
values of N would b e needed to resolve this question de nitively Unfortunately
such data are very time consuming to obtain b ecause the deterministic algorithm
2
for computing T and U takes a CPU time of order N Regarding the reverse
inequality it seems clear that p is strictly smaller than p
U T
In Table we rep ort our Monte Carlo estimates for dimensional self avoiding
walks at N Here we used the pivot algorithm combined
2
with deterministic algorithm For all observables we can get a reasonable
provided that we discard the data p oints at the two lowest values of N The ts to
2
hT i hT i and var T yield p Again the errors here are purely
T
statistical they do not take account of systematic errors arising from corrections to
scaling We will make a much more careful estimate of hT i and p in Section
T
The ts to hU i hV i and hV i yield p
B ar r ett K ar p Luby U
and resp ectively Again the agreement is medio cre and it is far
from clear whether p is strictly greater than p later we will see that
U T
2 2
p is a b etter estimate Otherwise put the ratio hT i N hU i
T N
decreases from at N to at N it is far from clear
whether this ratio is tending to zero or to a nonzero value as N Again data
at much larger values of N would b e useful
It is worth remarking that in b oth the dimensional and dimensional cases
2
var T scales the same way as hT i but is only ab out resp as big
This means that the probability distribution of T is very narrow Otherwise put
while N step SAWs vary radically among themselves in size and shap e the over
(1) (2)
lap T b etween two of them is remarkably constant Presumably this is
(1) (2)
b ecause the op eration of forming lls in the holes in the individual
(1) (2) (1) (2)
walks while and are fractals is semi solid roughly like Swiss
cheese
Numerical Results
Two Dimensions
2 2 2
In Table we present our data for hR i hR i hR i hT i and the pivot
N N N N N
e g m
algorithm acceptance fraction f for SAWs in dimension d in the range
N Most of these SAWs were generated using the pivot algorithm using
either dimerization or straight ro ds for initialization see App endix B for a discussion
6
of the adequacy of thermalization run lengths were b etween and
6
pivots subsequent to thermalization However some data at N were
5 5
generated by pure dimerization b etween and indep endent pairs of SAWs
(1) (2)
p er run The overlap T was in most cases estimated using the Karp Luby
algorithm Section with R however some runs at N used
deterministic algorithm Section Some subtleties concerning the correct
determination of error bars on data generated by the pivot algorithm are discussed
in App endix C
Early versions of our program had a bug in which SAWs were pivoted only at
sites k i e never at the starting p oint This is harmless for single SAWs
thanks to the lattice symmetries but for pairs of SAWs it means that the relative
orientation of the initial steps of the two SAWs is never altered by the algorithm
This causes a slight bias in the estimates of hT i esp ecially for small N However
we b elieve that this bias is completely negligible i e much less than our statistical
errors for the values of N treated here we have b een unable to detect any sys
tematic di erences b etween runs with and without this bug Therefore instead of
a
throwing away the tainted data we have simply indicated by an in Table those
values of N for which some not necessarily all of the data su ers from the bug
Caveat lector
2 2
Table shows the resulting values for the universal amplitude ratios hR i hR i
g e
2 2
i and The error bars are here determined using the triangle inequality i hR hR
e m
they are probably overestimates by a factor of
2 2 2
i hT i and f versus N are so straight that there i hR i hR Log log graphs of hR
m g e
is nothing to b e gained by repro ducing them here We t each of these quantities to
pow er
the Ansatz AN where p ower for the three squared radii for hT i
4
and p for f by p erforming weighted least squares regressions of their logarithms
against log N using the a priori error bars on the raw data p oints Table to
determine b oth the weights and the error bars Chapter As a precaution
against corrections to scaling we p erformed the t with a lower cuto N N
min
2
and we tried all p ossible values of N The value sum of squares of normalized
min
deviations from the regression line can serve as a test of go o dness of t Let us
2
de ne the signi cance level as the probability that would exceed the observed
value assuming the correctness of the p ower law mo del and of the raw data error
2
bars this probability can b e read o the distribution with D n degrees of
freedom DF where n is the number of data p oints used in the t An abnormally
2
large value of say a level less than may indicate either that the pure p ower
law Ansatz is incorrect e g due to corrections to scaling or else that the claimed
error bars on the raw data are to o small further investigation would b e necessary
18 2
to determine which of these is the true cause An abnormally small value of
say a level greater than probably indicates that the claimed error bars on
the raw data are to o large
2 2 2
The exp onents estimated from ts to hR i hR i hR i and hT i are plotted
e g m
as a function of N in Figure The exp onent estimated from the t to f is
min
2
plotted in Figure Lo oking at the values in these ts we nd statistically
2 2
signi cant corrections to scaling in hR i and hR i only for N in hT i only
min
e g
Using in each case the next larger and in f only for N for N
min min
value of N we nd
min
2
hR i
e
A
N
min
2
DF level
2
hR i
g
A
N
min
2
DF level
2
i hR
m
A
N
min
2
DF level
hT i
4
A
N
min
2
DF level
f p
A
N
min
2
DF level
error bars are one standard deviation The results are in excellent agreement with
the b elieved exact value although some very slight corrections
to scaling clearly remain and with the hyperscaling relation d
4
2 2 2
For hR i hR i hR i and hT i b etter estimates of the nonuniversal amplitude
e g m
A can b e obtained by imp osing the exp onents and simply
4
18 2
Note also that an abnormally large change in as N is increased by one step that is a
min
2
drop in by an amount signals that the data p oint in question di ers from the regression
line by several standard deviations This could indicate that the corrections to scaling at this value
2
of N are signi cant even though the overall which is dominated by contributions from larger
N may lo ok reasonable
3 2
tting observable N to a constant For all four observables this ratio declines
very slightly with N by N it is within error bars of its apparent asymptotic
value Taking N we obtain
min
2
hR i A
e
2
DF level
2
hR i A
g
2
DF level
2
i A hR
m
2
DF level
hT i A
2
DF level
3 2
Another approach is to t observable N to a b N with a b and all
19 2
variable For hR i for which we have data only at N we are unable to
m
obtain a decent t with any value of N available to us the corrections to scaling
min
2 2
i and hT i we obtain reasonable ts even i hR at N are to o weak For hR
g e
for N
min
2
i a hR
e
b
2
DF level
2
hR i a
g
b
2
DF level
hT i a
b
2
DF level
Of course the exp onents pro duced by the ab ove ts should not b e taken to o se
riously they could well b e phenomenological e ective exp onents that summarize
the combined e ects of two or more correction to scaling terms e g the leading
1
1
non analytic correction N plus the analytic correction N over some particu
lar range of N These ts are nevertheless useful in providing simple interpolation
19
The purp ose of writing here N instead of N is to reduce the correlation b etween the
estimates of b and Of course can equally well b e replaced by any reasonable value which is
roughly in the middle of the N values represented by the data p oints
formulae that summarize our data within error bars
2 3 2 0 657
hR i N N
e
2 3 2 0 557
hR i N N
g
3 2 0 581
hT i N N
In particular we can use these formulae to compare our data with other workers
data at di erent values of N but only for N
The upshot is that the corrections to scaling are quite weak in the two dimensional
self avoiding walk we can barely see them at our level of precision A serious
study of corrections to scaling in this mo del would therefore require much higher
statistics than are available here at large but not to o large values of N say
We have b egun such a study and will rep ort the results sepa N
rately
2 2 2 2
The corrections to scaling on the amplitude ratios hR i hR i hR i hR i and
g e m e
are even weaker than on the original observables this is b ecause the original correc
tions to scaling all have the same sign The two ratios of radii have no statistically
signi cant corrections to scaling at our level of precision even if one considers the
true error bars to b e of those rep orted in Table The interpenetration ratio
but these corrections are do es have noticeable corrections to scaling at N
very small see Figure Fitting the data for N N to a constant we obtain
min
2 2
i i hR hR
e g
N
min
2
DF level
2 2
i i hR hR
e m
N
min
2
DF level
N
min
2
DF level
The theorists interpenetration ratio de ned in is
R
Three Dimensions
2 2
In Table we present our data for hR i hR i hT i and the pivot algorithm
N N N N
e g
acceptance fraction f for SAWs in dimension d in the range N
Most of these SAWs were generated using the pivot algorithm using either
dimerization or straight ro ds for initialization see App endix B for a discussion of the
6 7
adequacy of thermalization run lengths were b etween and pivots
subsequent to thermalization However some data at N were generated
5 5
by pure dimerization b etween and indep endent pairs of SAWs p er
(1) (2)
run The overlap T was in most cases estimated using the Karp Luby
algorithm Section with R however some runs at N used
deterministic algorithm Section Data tainted by the bug concerning hT i
a
are again indicated by see Section
2 2
Table shows the results for the universal amplitude ratios hR i hR i and
g e
The error bars are determined using the triangle inequality they are probably
overestimates by a factor of
2 2 pow er
We b egan by tting hR i hR i hT i and f to the pure p ower law Ansatz AN
e g
2 2
The exp onents estimated from ts to hR i hR i and hT i are plotted as a function
e g
of N in Figure The exp onent estimated from the t to f is plotted in Figure
min
Very strong corrections to scaling are apparent for all these observables the
4
or more exp onent estimates do not app ear to stabilize until one takes N
min
Nevertheless the estimates of do app ear to b e converging to a common value
20
in particular hyperscaling app ears to b e satis ed
Another view of these same ts is presented in Figure here we plot the
0 5
estimate of versus N The idea here is that the correction to scaling exp onent
min
is predicted by RG calculations to b e in the vicinity of
1
see Section if this prediction is correct then each set of estimates should
fall roughly on a straight line at least asymptotically as N The data are
min
consistent with this prediction but the large uctuations make clear that it will b e
di cult to get accurate estimates of by Monte Carlo
1
pow er pow er 2 2
i and hT i to the Ansatz AN BN i hR Next we tried tting hR
g e
where p ower for the squared radii and for hT i we now assume hyperscaling
For each choice of xed exp onents and we determine A and B by weighted
2
least squares and record the resulting we then ask which pairs lead to
2
an acceptable Here acceptable is taken to mean a signi cance level
for D n degrees of freedom where n is the number of data p oints This
corresp onds roughly to con dence limits of one standard deviation The results
2 2
for hR i and hT i are shown in Figure The result for hR i is similar to that for
g e
2
hR i but the band of allowed values is wider For N there are no pairs
min
g
2
that are satisfactory for b oth hR i and hT i This means that at such small
g
N one needs more than a single correction to scaling term to t b oth observables in
a way compatible with hyperscaling and universality On the other hand for larger
N one obtains a swath of acceptable pairs all contained in the range
min
and From this analysis it seems di cult to
obtain much precision on but it do es suggest that
sub jective con dence limits
pow er
Now supp ose we imp ose and t observable N where p ower
for the squared radii and for hT i to a b N with a b and all
variable For N the estimates of from these three ts are not consistent
min
20
Note that the rigorous inequality d see Theorem A and equations A A
4
in App endix A means that the limiting value of the upp er pair of curves must b e ab ove or equal to
the limiting value of the lower curve
this might b e guessed from Figure However for N they are consistent
min
and yield
pow er pow er
If we now return to the Ansatz AN BN and take and
2 2
in hR i for N we obtain reasonable ts in hR i for N
min min
g e
Taking N we obtain and in hT i for N
min min
2
hR i A
e
B
2
DF level
2
hR i A
g
B
2
DF level
hT i A
B
2
DF level
At the very least these ts provide simple interpolation formulae that summarize
our data within error bars but for N only
1 1754 0 6154 2
i N N hR
e
2 1 1754 0 6154
hR i N N
g
1 7631 1 2031
hT i N N
If we take these ts seriously we obtain the results
(1) (1)
b b
R R
g e
(1) (1)
b b
A R
e
for the universal ratios of correction to scaling amplitudes
Another way to study the corrections to scaling is to lo ok at an amplitude
ratio which is known to tend to a nonzero constant as N and t it to
2 2
a b N with a b and all variable Two candidates are hR i hR i and
g e
2 2
assuming hyperscaling Unfortunately the corrections to scaling in hR i hR i
g e
are to o weak to yield much information on see Table the amplitude b is
within of zero and the error bars on are enormous However we can obtain
2 2
a reasonable estimate of the universal limiting value hR i hR i a Note rst that
g e
2 2 2
the unusually small con rms our b elief that the true error bars on hR i hR i
N N
g e
are roughly of those rep orted in Table If so then the error bars on a b and
in Table should also b e reduced by a factor of Making this adjustment
we conclude that
2
hR i
g
2
hR i
e
here the rst error bar represents systematic error due to uncontrolled corrections
to scaling sub jective con dence limits and the second error bar represents
the adjusted statistical error classical con dence limits
The analysis of is much more favorable for estimating as the corrections
to scaling are rather strong Figure This is b ecause the corrections to scaling
2
on hT i and hR i have opp osite signs as is evident from Figure A sample of
g
the results is shown in Table Note that the estimates of b and are quite
stable as N is increased and that the error bar on is remarkably small The
min
2
unusually small con rms our b elief that the true error bars on are roughly
of those rep orted in Table If so then the error bars on b and should
also b e reduced by a factor of The excellent t can b e seen graphically in
0 56
Figure is amazingly close to a linear function of N This con rms our
exp ectation that the correction to scaling exp onent is approximately and
1
suggests that it may b e slightly higher around A fair estimate would b e
1
sub jective con dence limits On the other hand the pro duced by the
ab ove t could well b e a phenomenological e ective exp onent that summarizes
the combined e ects of two or more correction to scaling terms e g the leading
1
1
plus the analytic correction N over some particular range of N correction N
The only way to sort this out would b e to use a larger N together with improved
min
statistics In any case we can estimate the universal limiting value
We have again made the factor of adjustment in the statistical error bar The
theorists interpenetration ratio de ned in is
R
Since hyperscaling is satis ed
Discussion
Comparison with Previous Numerical Studies
Two Dimensions
We rst compared our raw data with those of other studies
making direct comparisons where the values of N match and using the interpolation
21
formulae at other values of N We nd excellent agreement with one
p erplexing exception when compared with the extremely precise data of our
2 2
values for hR i and hR i are ab out low at N and slightly low
e g
at N Interestingly these are precisely the runs that were p erformed
21
Warning We b elieve that the theoretical premises of are erroneous in that the authors of
fail to distinguish correctly which quantities are universal and which are non universal See
Section ab ove Nevertheless the Monte Carlo data in are useful
using pure dimerization while the runs at N used the pivot algorithm It is
therefore conceivable that our dimerization program has a subtle bug which causes
a very small error p erhaps one that decreases with N However we have b een
unable to nd any such bug and the small discrepancy could well b e a statistical
uke
2 2 2 2
Our estimates of the universal ratios hR i hR i and hR i hR i agree p erfectly
g e m e
with the b est published estimates and have roughly the same precision In
particular they con rm the b eautiful conformal invariance prediction of Cardy and
Saleur as corrected by
2
2
hR i
hR i
g
m
2 2
hR hR i i
e e
Much more precise data will b e available so on
To our knowledge there are no previous series extrap olation or Monte Carlo
22
estimates of for two dimensional p olymers But we can compare our value
with eld theoretic estimates
Monte Carlo simple cubic lattice N this work
2
Edwards mo del through order naive sum App endix D b elow
2
Edwards mo del through order exploiting d value App endix D b elow
This last estimate is amazingly close to the correct value it would b e interesting to
know whether this close agreement is an accident
Three Dimensions
We rst compared our raw data with those of other studies
making direct comparisons where the values of N match and using the interpola
23
tion formulae at other values of N We nd excellent agreement We
also compared our raw data with Nickel s interpolation extrap olation formulae
equations and which are based on exact enumeration of short chains combined
2
with high precision Monte Carlo data at N The agreement for hR i and
e
2
hR i is excellent most of the p oints di er by less than ab out di er by
g
b etween and and none di er by more than However for hT i there are
some mo dest discrepancies for N and N our values are all b etween
and lower than Nickel s while for N ab out half of our values are
b etween and higher than Nickel s We do not know whether either of these
discrepancies is real or what might b e its cause We susp ect that it arises simply
from statistical errors in Nickel s raw data for hT i which might b e several times as
large as our errors these would induce statistical errors in the co e cients of his
22
Table see Section b elow gives the values of for N from exact enumeration
N
But this series is much to o short to b e usefully extrap olated
23 2 2
Warning Reference uses an unconventional de nition of hR i it is N times ours
g
Note also that the data of references lack error bars
extrap olation formula which would in turn induce correlated statistical errors at
nearby values of N
Next we compare our estimates of the critical exp onents and with previous
1
work
Monte Carlo simple cubic lattice N this work
1
Monte Carlo simple cubic and BCC lattices N
Monte Carlo simple cubic lattice N
Monte Carlo simple cubic lattice N
Series extrap olation various lattices
RG n eld theory
1
24
RG n eld theory
g
6 25
Edwards mo del through order z
1
6 25
Edwards mo del through order z
1
all Monte Carlo error bars are one standard deviation It is clear in retrosp ect
that the earlier Monte Carlo estimates based on shorter walks than
those used here were biased upwards due to corrections to scaling This e ect is
seen very clearly in Figure Moreover if we truncate our own data to lie in the
same range of N as the previous studies the resulting estimates of are almost
identical to the quoted ones We have now done what we b elieve is a careful analysis
of the corrections to scaling and we have obtained reasonably go o d control over
them We therefore think that the current estimate is correct within its claimed
error bar But we could b e wrong We do not know why the series extrap olation
estimates are also high but it could arise from the same e ect p erhaps the walks
prob ed in these analyses up to steps are simply not long enough to p ermit
an adequate analysis of corrections to scaling even using the most sophisticated
di erential approximants metho ds
24
The b est estimate of for g is Comparison of references and suggests that
the uncertainty in g may b e of order This would add an extra uncertainty of order
to
25 7
The term of order z has recently b een obtained but the extended series has not yet to
our knowledge b een analyzed
The amazing fact is that the Monte Carlo estimates of have stabilized at almost
exactly the value predicted by the eld theoretic calculations in their various equiv
alent forms n eld theory or Edwards mo del
The very high accuracy of the eld theoretic calculations is rather surprising to us
since this metho d is susceptible to serious and quite p ossibly undetectable sys
tematic errors arising from a con uent singularity at the RG xed p oint
The weakness of this e ect may b e related to the apparent fact that the
con uent exp onents and are b oth very close to an integer namely
1 2 1
they are
On the other hand for the con uent exp onent the agreement b etween our
1
Monte Carlo estimates and the eld theoretic predictions is not so go o d Our Monte
Carlo data for can b e t amazingly well all the way down to N by the
Ansatz a b N with statistical errors only This exp onent
di ers considerably from the eld theoretic predictions which all lie at
1
We do not know which of these estimates if either is the correct one indeed
there are go o d reasons to b e distrustful of b oth On the one hand our Monte Carlo
estimate could well b e a phenomenological e ective exp onent that summarizes
the combined e ects of two or more correction to scaling terms e g the leading
1
1
plus the analytic correction N over some particular range of correction N
N To test this we would need to go to larger N and obtain signi cantly
min
improved statistics As things stand our data do not rule out the eld theoretic
prediction provided that one includes suitable subleading correction
1
to scaling terms On the other hand the eld theoretic prediction for should
1
also b e taken with a grain of salt it arises from the slop e of the function at the
xed p oint g g and as Nickel showed long ago this function has in
addition to the desired term linear in g g also nonanalytic terms prop ortional to
1
2 1 1
and the like see Section b elow for further discussion g g g g
But the numerical metho ds currently employed to
extrap olate the p erturbation series from g to g g assume contrary to fact
p
that g is regular at g g The presence of terms g g with p could
well lead to systematic errors in estimates of the slop e g It is therefore
worth considering the p ossibility that the true is indeed closer to than
1
to and that the error bars on the eld theoretic estimates may b e over
optimistic
Let us also make a brief comparison with the exp erimental results on p olymers
in a go o d solvent a more detailed analysis will app ear elsewhere The most
systematic data on the static scaling b ehavior of high molecular weight p olymers in
a go o d solvent were obtained in the s by three Japanese groups
using p olystyrene in b enzene These data were reanalyzed shortly after the app ear
ance of the RG predictions for by Cotton After making corrections
for p olydisp ersity Cotton obtains the value in go o d agreement
with the RG prediction and now in go o d agreement with the Monte Carlo data as
well Unfortunately we b elieve that this exp erimental value is unreliable for the
following reasons
The samples of Yamamoto et al have an unknown p olydisp ersity but
this p olydisp ersity is certainly not zero as Cotton s analysis implicitly as
sumed
The measurements of Fukuda et al were apparently a icted by a serious
systematic error of magnitude ranging from to arising from the way
26
that the solution concentration was measured see fo otnote
The data from the three di erent lab oratories cover almost nonoverlapping
ranges of molecular weight As a result the combined analysis of the three
sets of data is highly susceptible to the e ects of small systematic discrepancies
in absolute calibration b etween the three lab oratories as well as to the fact
that the three lab oratories used slightly di erent temp eratures C for
and C for
The raw data lack error bars As a result it is imp ossible to distinguish cor
rections to scaling or systematic exp erimental errors from statistical errors
This distinction is crucial to extracting a reliable value for as our analysis
of Monte Carlo data has demonstrated
A new generation of exp eriments using mo dern ultra sensitive light scattering in
strumentation and an optimal statistical analysis could prove
to b e very exciting
2 2
Next let us compare our estimates of the universal ratio hR i hR i with previous
g e
work
Monte Carlo simple cubic lattice N this work
Monte Carlo simple cubic lattice N
Monte Carlo b o dy centered cubic lattice N
Monte Carlo simple cubic lattice N
Monte Carlo simple cubic lattice N
Monte Carlo simple cubic lattice N
4
Edwards mo del through order z
27
all Monte Carlo error bars are one standard deviation The agreement is excellent
2 2
It is worth noting that hR i hR i is non monotonic as a function of N for small
g e
2 2
exact enumeration shows that hR i hR i is considerably above N
g e
our data in N its limiting value but for larger N e g
2 2
Table show unequivocally that hR i hR i is below its limiting value and our t
g e
26
Unfortunately it will not su ce simply to replace the measurements of Fukuda et al by
those of Utiyama et al on a subset of their samples b ecause these latter measurements may
su er from a systematic error of their own arising from the extrap olation to zero scattering angle
see fo otnote
27
There also exists a calculation in the Edwards mo del to second order in d but
unfortunately this expansion is to o ill b ehaved to b e extrap olated reliably to see p
Table suggests that as N the approach to the limiting value is also from
below It would b e interesting to know whether series extrap olation techniques can
sense this nonmonotonicity which lies b eyond the present enumerations and
predict approximately the correct limiting value
Next let us compare our estimates of the limiting interpenetration ratio with
previous work
Monte Carlo simple cubic lattice N this work
Monte Carlo simple cubic lattice N
2
Edwards mo del through order z Pad e App endix D b elow
2
Edwards mo del through order naive sum App endix D b elow
2
Edwards mo del through order exploiting d App endix D b elow
28
all Monte Carlo error bars are one standard deviation Our estimate is thus
consistent with the very recent Monte Carlo estimate of Nickel et al but
is ab out times as precise The Edwards model renormalization group estimates
are all of the right order of magnitude but most of them are not terribly close to the
correct answer This is hardly surprising in view of the very short p erturbation
series on which these estimates are based The expansion result augmented by
exact information at d is however amazingly close to the correct value b oth
in d and d It would b e useful to obtain a b etter understanding of whether
3
this is a coincidence or not p erhaps by calculating the O term in
Finally it is worth noting that the exp erimental values for also lie in the range
Section F Sections and or more optimistically
they average to However the exp erimental measurements of
are sub ject to all the problems noted earlier as well as the danger of addi
tional serious systematic errors arising from curvature in the extrap olation to zero
concentration This is a particularly severe problem for older studies which used
higher concentrations in order to comp ensate for the less sensitive light scattering
instrumentation then available Again new exp eriments would b e highly desirable
As explained in Section also the ratios of correction to scaling amplitudes
(1) (1) (1) (1)
b b and b b are universal So ideally we would like to measure these am
R R A R
g e e
(1) (1)
plitude ratios and compare them with the Edwards model prediction b b
R R
g e
Unfortunately the correction to scaling amplitudes are ex
tremely sensitive to the choice of exp onents and Therefore the only sensible
1
comparison that can b e made is to impose the same values of and as are
1
used in namely and and then p erform a weighted
1
least squares t to estimate the amplitudes Unfortunately the resulting estimates
of the correction to scaling amplitudes are highly unstable as a function of N If
min
(1) (1)
we take N we obtain b and b hence
min
R R
e g
(1) (1)
b b not a very useful estimate
R R
g e
28
Note also that Table see Section b elow gives the values of for N from exact
N
enumeration But this series is much to o short to b e usefully extrap olated
Finally we can compare the estimates of the pivot algorithm acceptance fraction
exp onent p
Monte Carlo simple cubic lattice N this work p or p erhaps lower
Monte Carlo simple cubic lattice N p
Monte Carlo tetrahedral lattice N p
Monte Carlo simple cubic lattice N p
There is a slight discrepancy b etween our results and the very careful work of Zi erer
we do not understand its origin It is conceivable though in our opinion un
likely that the critical exp onent for the pivot algorithm acceptance fraction might
vary from one three dimensional lattice to another More likely the magnitude and
even the sign of the corrections to scaling may vary radically b etween lattices
The Sign of Approach to
For several decades most work on the b ehavior of long chain p olymer molecules
in dilute solution has b een based on the two parameter the
29
ory in one or another of its variants traditional Flory type pseudo traditional
30 31
mo di ed Flory type or mo dern continuous chain type All two parameter
theories predict that in the limit of zero concentration the mean square end to end
2 2
i and the interpenetration ra i the mean square radius of gyration hR distance hR
g e
tio dep end on the degree of p olymerization N or equivalently on the molecular
weight M NM according to
monomer
2
i AN F hR bN a
R
e
e
2
hR i AN F bN b
R
g
g
F bN c
where F F F are claimed to b e universal functions which each sp eci c two
R R
e g
parameter theory should predict and A and b are non universal scale factors de
p ending on the p olymer solvent and temp erature but indep endent of N The
2 2 d 2 d 2
conventional notation is F F F z and z bN in h
R R
e g
R S S
spatial dimension d Moreover virtually all the theories and in particular the
mo dern continuous chain based theories predict that F is a monotone increas
ing and concave function of its argument bN which approaches a limiting value
for d as bN
29
See Yamakawa Sections and pp and and parts of Sections b
and b pp and See also DesCloizeaux and Jannink Section
pp
30
See Yamakawa most of Section pp and parts of Sections b and b pp
and See also Domb and Barrett
31
These theories take as their starting p oint the Edwards mo del of a weakly self avoiding contin
4
uous chain The Edwards mo del is also equivalent to the continuum eld
theory with n comp onents See DesCloizeaux and Jannink for a detailed treatment of the
Edwards mo del
But our Monte Carlo data for the self avoiding walk Figures and show
precisely the opp osite b ehavior is a decreasing and convex function of N which
approaches a limiting value d or d as N
The same b ehavior was found by Nickel The decrease of with N is strong
in d and weak but noticeable in d
In retrosp ect this b ehavior is heuristically almost obvious Short self avoiding
walks b ehave roughly like hard spheres i e is on the same order of magnitude
as the hard sphere value see Table On the other hand long self avoiding
walks are fractal ob jects i e thinner than hard spheres so one exp ects
This is manifestly true in dimension d where and in
har d spher e
2
dimension d where O it is natural to exp ect and we now
con rm that it is true also in d and to a lesser extent in d Of course
32
in d SAWs are hard spheres i e hard ro ds so The
har d spher e
monotonic decrease of as a function of d is shown in the last line
har d spher e
of Table If one now conjectures the simplest b ehavior namely that is a
N
monotonic function of N it follows that in dimension d must approach
N
its limiting value from above
There is also experimental evidence that for real p olymers in a
su ciently go o d solvent the approach to is from ab ove contrary to the two
parameter theory This b ehavior was considered to b e a p erplexing anomalous
e ect and various purp orted explanations were advanced
The correct explanation in our opinion was given three years ago by Nickel
see also theories of two parameter type are simply wrong Indeed
they are wrong not merely b ecause they make incorrect predictions but for a more
fundamental reason they purp ort to make universal predictions for quantities that
33
are not in fact universal Two parameter theories predict among other things
2 2 2
in particu i i hR hR that is a universal function of the expansion factor
T
g g S
lar is claimed to dep end on molecular weight and temp erature only through the
2
particular combination M T This prediction is quite simply incorrect b oth
S
for mo del systems and for real p olymers Indeed even the sign of the deviation
from the limiting value is not universal
All this has a very simple renormalization group explanation so it is
surprising that it was not noticed earlier As mentioned already in Section stan
dard RG arguments predict for any real or mo del p olymer chain the asymptotic
b ehavior
(1)
2 2
1
hR i A N b N a
R
e
e R
e
(1)
2 2
1
hR i A N b N b
R
g
g R
g
(1)
1
b N c
32
In d is essentially constant with N it di ers from only by corrections of order N
N
p
p
1
arising from the discreteness of the chain The exact formula is N N N
N
2
33
More precisely two parameter theories in which the scale factor A is indep endent of temp erature
In these are termed strong two parameter theories
as N at xed temp erature T T The critical exp onents and are
1
(1) (1) (1)
universal The amplitudes A A b b b are nonuniversal in fact even the
R R
e g
R R
e g
(1) (1) (1)
signs of the correction to scaling amplitudes b b b are nonuniversal However
R R
e g
the RG theory also predicts that the dimensionless amplitude ratios A A
R R
g e
(1) (1) (1) (1)
are universal and b b b b
R R R
e e g
Recently however several pap ers have app eared which at
tempt to explain the observed approach to from ab ove in terms of either an al
leged second branch of the renormalized eld theory with renormalized coupling
constant g the xed p oint value g or an extended two parameter
theory containing an extra parameter We confess that we have b een
unable to understand the conceptual basis of these pap ers Here is our attempt
to clarify what is going on
We b elieve that critical phenomena are b est understo o d in a Wilson type or
Wilson de Gennes type renormalization group framework By this we
mean an RG map R acting on the in nite dimensional space of Hamiltonians
for a eld theory or p olymer mo del with some xed ultraviolet cuto
e g living on a lattice the map R acts by integrating out the high momentum
34
short wavelength degrees of freedom We wish to contrast this with a eld
theoretic RG which acts in a nite dimensional space of continuum
renormalized eld theories or p olymer mo dels i e mo dels with the ultraviolet
cuto already taken to in nity by spatial dilation A few paragraphs from now we
will explain the eld theoretic RG in terms of the Wilson RG and we will see that
the reverse is not p ossible But for the moment let us simply pro ceed with the
Wilson approach
Figure taken from shows part of the Wilson renormalization group
ow for a cuto eld theory or p olymer mo del in dimension d Here
H is the Gaussian xed p oint which for d is also the xed p oint con
is the nontrivial go o d solvent trolling the theta solvent b ehavior while H
GS
Wilson Fisher xed p oint Please ignore for now the curves M M and C
s u
More precisely Figure shows the ow on the critical surface corresp onding to
correlation length in the cuto eld theory or chain length N in the
p olymer mo del Non critical mo dels i e or N lie ab ove the plane
of the page Both H and H have unstable relevant directions coming out
GS
of the page and these tra jectories lead to the in nite temp erature xed p oint H
which has or N
We must now distinguish two very di erent limiting situations in p olymer the
ory
I N at xed temp erature T where either a T T b T T or c
T T
34
For example for a continuous space eld theory with ultraviolet cuto the map R could b e
integration over eld comp onents with momenta in the shell jpj for a lattice eld
theory R could b e passage to blo ck spins for a discrete p olymer chain R could b e decimation
of o dd numbered sites along the chain
I I N T T with x N T T xed where is a suitable crossover
exponent
Roughly sp eaking case Ia corresp onds to the go o d solvent regime while case II
35
corresp onds to the crossover regime near the theta p oint
Case Ia applies to any one parameter family F of Hamiltonians parametrized by
the chain length N that transversally intersects the critical surface i e the plane
of the page at some p oint within the domain of attraction of H i e anywhere to
GS
the right of M For example the family F could intersect the critical surface at
s
P Q or R among many other places Then the critical exp onents and universal
amplitude ratios cf asso ciated with the limit N in the family
F are completely determined by the RG ow in an in nitesimal neighborho o d of
H they are therefore the same for al l families intersecting the critical surface
GS
within the domain of attraction of H universality On the other hand the
GS
nonuniversal amplitudes arise from the entire history of the RG ow that takes F
they are therefore di erent for di erent families F For example a family to H
GS
crossing the critical surface at P or R would approach from b elow while a family
crossing at Q would approach from ab ove The nonuniversal amplitudes cannot
b e predicted except through detailed knowledge of the microscopic physics i e of
the family F In particular they cannot b e predicted by any coarse grained
theory such as a renormalized eld theory or indeed by any simple mathematical
mo del
As will b e explained b elow the manifold M which extends also out of the
u
plane of the page corresp onds to the continuum Edwards mo dels with the plane
of the page corresp onding to z It is imp ortant to note that M has no special
u
status with resp ect to the go o d solvent xed p oint H it is merely one of many
GS
The tra jectories whose intersection with the critical surface is attracted to H
GS
universal prop erties of p olymer chains in a go o d solvent case Ia can indeed b e
extracted from the limit z of the continuum Edwards mo del but they can b e
extracted equally well from the limit N of any family F whose intersection
with the critical surface is attracted to H One family may b e more convenient
GS
for computation than another but all have the same conceptual status
Let us now explain the relation b etween the Wilson RG and continuum eld
theories Let H b e any critical xed p oint for the moment it do esn t matter
whether H is Gaussian and let M resp M b e the stable resp unstable
s u
36
manifold of H Continuum limits are obtained by taking a sequence of initial
Hamiltonians H approaching the stable manifold and rescaling lengths by suitable
n
factors This rescaling is equivalent to applying the map R a suitable number
35
However it is crucial to understand that cases I and I I refer to families of limiting paths in the
T N plane not to regions or domains of the nite T N plane Failure to appreciate this
distinction which may at rst seem rather p edantic can lead to apparent paradoxes regarding
the sign of the correction to scaling terms
36
For simplicity let us assume that there are no marginal op erators The presence of marginal
6
op erators such as the op erator at the Gaussian xed p oint in d do es not a ect the
fundamental conclusions but merely induces multiplicative logarithmic corrections
e n
of times The low energy e ective Hamiltonians H R H then tend to the
n
n
unstable manifold see Figure Continuum eld theories F are thus in one to
one corresp ondence with Hamiltonians H on the unstable manifold the correlation
functions of F at momenta jpj are equal to the correlation functions of H with
cuto This p oint of view has b een emphasized by Wilson and Kogut and
others
In particular the Gaussian xed p oint H has in dimension d a
two dimensional unstable manifold M the two unstable directions corresp ond to
u
2 4
and interactions or in p olymer language a chain length fugacity and a two
37
b o dy self interaction So M corresp onds to the manifold of sup errenormalizable
u
4
continuum eld theories or in p olymer language to the continuum Edwards
mo del
We can now understand the connection b etween the Wilson and eld theoretic
38
renormalization groups The homogeneous eld theoretic RG equations
describ e how a family of continuum eld theories is mapp ed into itself under spatial
dilation On the other hand the continuum eld theories are in one to one corre
sp ondence with Hamiltonians on the unstable manifold and this corresp ondence
takes spatial dilation into the RG map R Thus the eld theoretic RG is nothing
other than the Wilson RG restricted to the unstable manifold and then rewritten in
terms of renormalized parameters
Having understo o d this we can now evaluate the attempts to
explain the wrong sign of approach to or the analogue in liquid gas critical
p oints in terms of an alleged second branch of the renormalized eld theory
39
lo cated at g g The trouble is that as far as we know no such branch exists
As explained ab ove continuum eld theories corresp ond to the unstable manifolds of
critical xed p oints Thus the alleged second branch could exist only if there were
lying somewhere to the right of H an as yet unknown critical xed p oint H
GS new
part of whose unstable manifold is attracted to H There is to our knowledge
GS
40
no evidence whatso ever for the existence of such a xed p oint
37 2
We ignore here the marginal op erator r which corresp onds to a physically trivial rescaling
of eld strengths
38
This connection was worked out by Kupiainen and Sokal in and was quite p ossibly known
to others as well However to our knowledge it rst app eared publicly in Very similar ideas
app eared earlier in work of Hughes and Liu
39
Of course no such explanation is needed b ecause we have already given a complete and straight
forward explanation in terms of the Wilson RG But there would b e no harm in giving an alternate
explanation of the same phenomenon provided that this explanation is correct
40
The nonexistence of such a branch of continuum eld theories seems to b e recognized by Kr uger
and Sch afer who note abstract and p that the strong coupling branch implicitly relies
on the existence of a nite segment size ultraviolet cuto But on the other hand they insist
that renormalized eld theory can b e used for g g p We do not understand how
these two statements can b e reconciled Perhaps Kr ugerand Sch aferwant to study the cuto eld
theories to the right of H But there is no distinguished one parameter family of such theories i e
GS
the putative extension of M there is simply the in nite dimensional space of al l cuto theories
u
with complicated Hamiltonians as well as simple ones lying to the right of H all of which have
GS
The conventional eld theoretic RG approach makes another
error in assuming that the function describing the eld theoretic RG ow is reg
ular at g g In fact as p ointed out long ago by Nickel the function
1
1 2 1
should b e exp ected to contain nonanalytic terms like g g g g and
many others This can easily b e understo o d in our Wilson type RG framework The
eld theoretic function describ es the Wilson RG ow restricted to the unstable
manifold M and rewritten in terms of the renormalized parameter g Now M
u u
41
has no sp ecial status at H like every other RG tra jectory it approaches H
GS GS
tangent to the leading irrelevant direction but barring a miraculous coincidence
M also has nonzero comp onents with resp ect to the nonlinear scaling elds at
u
H in all of the subleading irrelevant directions This induces nonanalytic terms
GS
2 1 3 1
g g g g in the function In addition the analytic correc
1 2
1 1
tions to scaling at H induce nonanalytic terms like g g g g
GS
in the function
In summary the eld theoretic function is nonanalytic as g g and is as
far as we know not de ned at al l for g g These facts cannot b e understo o d by
manipulation of formal expressions within renormalized eld theory but they can
b e understo o d when the renormalized eld theory is placed within the Wilson type
RG context
Finally this framework allows us to understand the sp ecial role played
but it by the continuum Edwards mo del This mo del has no sp ecial status at H
GS
do es have sp ecial status at H it is the unstable manifold M For this reason the
u
continuum Edwards mo del describ es the universal crossover scaling behavior in an
in nitesimal region just above the theta temperature namely the limit N T
T with x N T T xed case I I ab ove where d for d
4 11
1
42
and log for d That is the continuum Edwards mo del controls
2
the b ehavior of any two parameter family F of Hamiltonians that transversally
intersects the critical surface in some curve C which in turn transversally intersects
M see Figure Since the theta p oint is b eyond the scop e of the present pap er
s
we refer the reader to for details
equivalent conceptual status
The same ob jection applies to the extended two parameter theory of Chen and No olandi
Perhaps their extra parameter is intended to corresp ond to the co e cient of the leading
irrelevant coupling at H but in that case it is merely an inordinately complicated restatement of
GS
and moreover it neglects the second and higher irrelevant couplings Of course one could
consider the manifold corresp onding to the vanishing of the second and higher irrelevant couplings
with resp ect to the nonlinear scaling elds at H but this manifold on the left side of H
GS GS
do es not coincide with M
u
Finally we b elieve that the same ob jection applies to the second half of Nickel s pap er after
p in connection with his recursion mo del He seems to realize this as he describ es the
upp er branch in his simpli ed recursion mo del as quasi universal
41
Except for a measure zero set of exceptional tra jectories
42
By the latter expression we mean that in d the correct scaling variable is x
1 2 4 11
N log N T T
Prosp ects for Future Work
Let us close this pap er by mentioning brie y some interesting areas for future
work
Higher virial co e cients can b e studied by the same metho ds that we have used
here to study the second virial co e cient For example the third virial co e cient
b etween molecules of types i j k is
X X
(ij k )
W s W s W s B
i j k
3
0 d
s S x Z
i
0 00 d
s S x Z
j
00
s S
k
i i h i h h
0 0 00 00 00 00 0 0
V ((x s ) (x s )) V ((0 s) (x s )) V ((0 s) (x s ))
ij
j k ik
e e e
compare where the interaction energy V is given as usual by The
2
(N N N ) (N N )
ratio B B is exp ected to b e universal for p olymers in a go o d solvent
3 2
and it would b e interesting to know its value At present only a crude eld
2
theoretic estimate is available B B based on rst order p erturbation
3
2
43
theory It should b e noted that this quantity plays a crucial role in the extraction
of the second virial co e cient from exp erimental light scattering data due to the
necessity of extrap olating to zero concentration
The hit or miss algorithm Section can easily b e generalized to compute
th
the n virial co e cient given n indep endent SAW samples Moreover the e
ciency should remain reasonably go o d whenever hyperscaling holds We do not
know whether the Barrett and Karp Luby algorithms can b e generalized to virial
co e cients of order n Finally the deterministic Fourier metho d Section
can b e applied to the third virial co e cient and to at least some of the graphs for
virial co e cients of order n namely those graphs that can b e decomp osed
into series and parallel connections For example to compute B one rst uses the
3
Fourier metho d to compute x and thus I x and likewise for and one
12 12
then computes I I I by passing to Fourier space
12 13 23
Another interesting extension of our work would b e to study the Domb Joyce
mo del of weakly self rep elling walks in which each self intersection is p enalized
by a weight e obviously this mo del interpolates b etween ordinary
random walks and SAWs This mo del can b e studied in two
very di erent limits
Ia N at xed
2 d 2
I I N with x N xed
Case Ia is the go o d solvent regime and the universal quantities should take exactly
the same values as for SAWs for any On the other hand the non universal
43 2
See DesCloizeaux Jannink p equation Note that in their notation B B
3
2
4
2
h z g z compare their equation
3
quantities will manifestly b e dep endent we exp ect the approach to to b e from
b elow for small and from ab ove for large In particular there will b e some
intermediate value for which the leading correction to scaling will vanish In
Figure this would b e achieved e g by a family of Hamiltonians that crosses the
critical surface ab out halfway b etween P and Q Of course subleading corrections
2
to scaling will still b e present but these will b e suppressed by N which decays
1
much more rapidly than N By studying the Domb Joyce mo del systematically
in the N plane it should b e p ossible to lo cate empirically and to exploit this
knowledge to obtain estimates of and other universal quantities that are less
contaminated by the e ects of corrections to scaling We thank Jim Barrett and
Bernie Nickel for this observation This is in the same spirit as the analyses of
series expansions by Chen Fisher and Nickel who found that by varying an
irrelevant parameter and imp osing universality they could reduce the sensitivity of
exp onent estimates to the e ects of corrections to scaling
Case I I in the Domb Joyce mo del should b e given rather trivially by the con
tinuum Edwards mo del This fact is hardly in doubt but the results could serve
as a useful nonp erturbative check on the reliability of the extrap olations to large z
in the Edwards mo del This is a warm up for the problem of the crossover
scaling b ehavior near the theta p oint
The deep est and most di cult extension of this work would b e to SAWs
with nearest neighbor attraction for the purp ose of studying the crossover scaling
b ehavior near the theta p oint In dimension d the crossover scaling functions
are predicted to b e given exactly mo dulo two nonuniversal scale factors by
the continuum Edwards mo del In dimension d there is as yet no theoretical
prediction for the crossover scaling functions but there are some predictions for
critical exp onents It is a highly nontrivial problem to develop Monte Carlo
algorithms that work well near the theta p oint The pivot algorithm taken alone
do es not work terribly well in this regime
A Some Geometrical Theorems
In this app endix we prove some geometrical b ounds on T A B A B and
U A B see and that played a role in Sections and of this pap er
When averaged over pairs of indep endent SAWs of length N these b ounds show
that
d
a hT i const N
2
b hT i const N
c hU i log N
mo dulo the usual assumptions on the scaling of individual SAWs In particular
b ounds a and b together prove mo dulo these assumptions hyperscaling for
SAWs in dimension d
A Theorems and Pro ofs
d
Let us b egin by de ning several measures of the size of a set A Z
th
The span in the co ordinate direction
S A max e x min e x A
x A x A
where e is the unit vector along the axis d
The Euclidean diameter
diam A max kx x k A
2
0
x x A
where k k denotes the Euclidean norm
2
The sup norm diameter
diam A max kx x k max S A A
0
1 d
x x A
where kxk max je xj
1 d
d
Let R b e a p ositive real number A set A Z is said to b e R connected if for
all pairs x x A there exists a sequence x x x x x of p oints in A such
0 1 n
that kx x k R for i n A set is said to b e connected if it is connected
i i 1
Note that this allows diagonal as well as nearest neighbor connections
Finally we let R b e the maximum number of lattice p oints in a closed
d
Euclidean ball of radius R namely
d
fx Z kx x k Rg A R sup
0 2 d
d
x R
0
d d 1 d 2
Note that R R O R as R where d d is the
d d d
d
volume of the unit ball in R
d
Theorem A Let A B Z Then
d
h i
Y
S A S B A A B
=1
A B diam A diam B A
d
d
Theorem A Let A B be R connected subsets of Z d Let G O d be
d
d
d
the orthogonal symmetry group of Z so that G d Then
d
X
diam A diam B A A g B
2
G R
d
g G
d
d
Theorem A Let A B Z Then
N 1
min
X
N N
max min
U A B A a
k N
min
k =1
log N log N A b
max min
where N min A B and N max A B The rst inequality
min max
is the best possible in terms of N and N
min max
Proof of Theorem A These b ounds follow immediately from the fact that
the set A B can b e contained in a rectangular parallelopip ed or a sphere of the
indicated size
Proof of Theorem A Let us consider the case d rst Fix g G
2
There exist I and x x A such that jjx x jj R for i I
1 I i i 1
and jjx x jj diam A Similarly there exist J and y y g B such
I 1 1 J
that jjy y jj R for j J and jjy y jj diam g B diam B
j j 1 J 1
Then Lemma of Aizenman implies that
jjx x jj jjy y jj j sin x x y y j A g B
I 1 2 J 1 2 I 1 J 1
2
R
diam A diam B j sin x x y y j A
I 1 J 1
2
R
where v w is the angle that the vector w makes with resp ect to the vector v
Let g G b e the op erator of rotation by Then for any vectors v and
+ 2
w v g w v w so
+
j sin v w j j sin v g w j j sin v w j j cos v w j A
+
From A we see that
A g B A g g B diam A diam B A
+
2
R
The theorem for d now follows by averaging A over all g G and dividing
2
b oth sides by
Now consider the case d Fix g G Cho ose indices and so that
d
0
g B diam B If then let P b e the S A diam A and S
d
orthogonal pro jection of Z onto the co ordinate plane If then let
b e any element of f dg n f g and let P b e the orthogonal pro jection onto
the co ordinate plane Let h G b e the rotation by in the or
d
co ordinate plane which leaves all the other co ordinates xed Note that
P h g P where g G was de ned ab ove We now use the result A for
+ + 2
d
A g B A hg B PA P g B PA P hg B
diam PA diam P g B
2
R
diam A diam B A
2
R
The theorem now follows by averaging as for the d case
We remark that some strengthened versions of Theorem A while quite plausi
ble at rst sight are wrong For example we at rst thought that for connected
2
sets and in particular walks in Z
A B const S A S B A
1 2
This is false since if A and B are b oth N step self avoiding walks from to
N N that stay in the diagonal strip jx x j then A B N while
1 2
2
S A S A N Thus one must average over rotations in some way We had also
1 2
2
conjectured that for any sets A B Z
A B A g B C Pro j A Pro j B A
+
1 2
where Pro j is the pro jection onto the x co ordinate axis and g is rotation by
+
here C is a constant that is indep endent of the connectedness R A counterexample
to this conjecture was found for us by G L O Brien as follows Let M b e a large
integer and let
A B f k M l M k l M g A
2
This is a large sparse square slightly tilted Then Pro j A Pro j B M
1 2
2
but A B A g B M We still do not know if A holds if its
+
left hand side is replaced by the left hand side of A with d
Proof of Theorem A Throughout this pro of we shall assume without loss
of generality that N A N B
1 2
To show that the rst inequality is the b est p ossible let A resp B consist N
1
resp N consecutive sites on the x axis so A and B are parallel ro ds Then it
2 1
is easy to check that the rst inequality holds as an equality
d
Let v b e a vector in R such that the inner pro duct mapping x v x is
one to one on A B For example the co ordinates of v could b e d irrational
numbers that are linearly indep endent over the rationals Denote the elements of
A by a a ordered so that v a v a for every i Similarly denote the
1 N i i+1
1
elements of B by b b ordered so that v b v b for every j
1 N j j +1
2
We shall use the following observation
Observation If a b a b for some i l m then l i
i N l m
2
To prove this note that the hypothesis implies that a a b b and so
i l N m
2
v a v a v b v b Therefore v a v a which implies l i The
i l N m i l
2
pro of of the following observation is exactly analogous
Observation If a b a b for some j l m then l j
N j m l
1
We now claim that for any i
a b i A
i N
2
To prove this observe that x equals the number of distinct a s in A such that
l
x a B But if a b a B then l i by Observation This proves
l i N l
2
A Next we can use Observation in an analogous way to prove that for any
j
b j A a
j N
1
Now de ne the following subsets of A B
i N g A a S fa b
2 i N
2
b j N g A b S fa
j 2 N
1
k N N g A c S fa b
2 1 k N
2
It follows from Observation that S and S are disjoint and from Observation
that S and S are disjoint trivially S and S are disjoint Therefore
X X X
U A B
x x x
00 000 0
x S x S x S
N 1
2
X
N N
1 2
A
k N
2
k =1
where the last inequality uses A A and the trivial b ound x N A
2
n
P
1
k log n simple comparison of Riemann sum to integral shows that
k =1
A Application to SAWs
Now let us apply these b ounds to the case in which A and B are indep endent
(1) (2)
SAWs of length N We rst note that that the spans S S and the
1 d
diameters D and D of either one of these SAWs are exp ected to scale like N in
the sense that
k
k m (k +...+k + +m)
d
1 1
d
hS S D D i N A
1 d
for any exp onents k k m In particular this holds if the probability
1 d
distribution of the SAW with lengths rescaled by N converges weakly to some
probability measure on a space of continuum chains with resp ect to which the spans
have nite nonzero moments
Assuming such scaling it follows that is b ounded as N Proof By
N
Theorem A we have
d
h i
Y
(1) (2) (1) (2)
S S A T
=1
but from this and A it easily follows that
c
N N
(1) (2) d
hT i const N A
N N
2
c
N
Q E D Assuming the usual scaling for the radius of gyration it follows that
is b ounded as N
N
Equation A also implies that
(1) (2) 2
hT i const N A
N N
in any dimension d This may b e deduced from Theorem A as follows Since
S the set of all N step SAWs which start at the origin is invariant under lattice
N
symmetries we have
X
(1) (2) (1) (2)
hT i
N N
2
c
N
(1) (2)
S
N
X X
(1) (2)
g
2
G c
d
N (1) (2)
g G
S
d
N
X
(1) (2)
diam diam
2
c
N (1) (2)
S
N
2
A hD i
N
and combining this with A gives A
Lastly we observe that the scaling assumption A implies that hyperscaling
holds in two dimensions Indeed equations A and A together imply that
(1) (2) 2
hT i const N when d A
N N
B Adequacy of Thermalization in the Pivot Al
gorithm
As p ointed out in Section the initialization of the pivot algorithm is a highly
nontrivial issue For not to o large N up to a few thousand at least one can use the
dimerization algorithm to pro duce a p erfect equilibrium start However
for very large N dimerization is unfeasible and it is necessary to thermalize the
system by discarding the rst n N f iterations But this is painful
disc exp
b ecause is a factor N larger than for global observables A and for very
exp int A
5
large N the CPU time of the algorithm could end up b eing dominated by
the thermalization One is therefore tempted to cut corners in the choice of n
disc
In this app endix we want to illustrate how acquiescing to this temptation can lead
to disaster in the form of large systematic errors
Figure shows the temp oral history of our pivot algorithm run for the
dimensional SAW at N The initial con guration was a pair of parallel
2 5
ro ds We averaged the observables R and T over bins of pivot iterations each
e
and plotted the resulting averages for the rst bins Clearly severe initialization
2 5 5
bias is present in R until at least time and in T until at least time
e
Here we used the following seat of the pants criterion if the value at bin n is
larger than all subsequent bins and smaller than all preceding bins then severe
initialization bias is present at time n Signi cant initialization bias enough to
cause a systematic error comparable to the very small statistical error could
well b e present at times twice or three times this So to b e safe we discarded in
6 6
this case the rst iterations The total run length was approximately
iterations so a little less than half of the run was discarded
In general we exp ect the thermalization to require a time prop ortional to
exp
N f where f is the pivot algorithm acceptance fraction From the run shown
in Figure we infer that severe initialization bias is present until at least time
N f Therefore to b e safe we have always discarded at least N f iterations
except of course for runs using a dimerized start We b elieve that this rule is
su ciently conservative to render the systematic errors arising from inadequate
thermalization much smaller than the statistical errors
C Some Statistical Subtleties
Let A A b e the time series for some observable A in equilibrium As
1 n
P
n
1
discussed in App endix C the error bar on the sample mean A n A
i
i=1
is
1 2
C
int A AA
C stddev A
n
where
2
C t hA A i hAi C
AA s s+t
t C t C C
AA AA AA
X
t C
int A AA
t=
Since and C are not known they must b e estimated from the time series
int A AA
we de ne
n jtj
X
b
b
A A A A C C t
i AA
i+jtj
n jtj
i=1
b b
b b b
b
t C t C C
AA AA
AA
M
X
b b
b b
t C
int A
AA
t= M
where we must still choose the window M
One reasonable approach to choosing M is the automatic windowing algorithm
b
b
of App endix C choose M to b e the smallest integer such that M c M
int A
for a suitable window factor c If t were roughly a pure exp onential then it
AA
5
would su ce to take e g c since e However for global observables
A in the pivot algorithm t is exp ected to b e very slowly decaying after a
AA
1
brief initial decay one exp ects t t up until a time of order N f
AA exp
after which time t decays rapidly This is the b ehavior exhibited by the ex
AA
act solution of the pivot algorithm for ordinary random walk where t t
AA
in the intermediate region Section We con rmed this b ehavior empir
8
ically for the self avoiding walk by making an extremely long n pivots
b
b
simulation at N d the sample auto correlation functions t for
AA
2 2 2
there T are shown in a log log plot in Figure For R R A R
K ar p Luby R=20
e g e
is a wide intermediate region t in which the log log plot is roughly
q 2
there is more curvature straight yielding t t with q For R
AA
g
but the slop e q is in the same ballpark For T in contrast the curva
K ar p Luby R=20
q
ture is so great that we are unable to identify an intermediate region with clear t
b ehavior In any case we have rough empirical con rmation for the theoretically
predicted b ehavior we are unable to say whether q or q p or something
else but clearly q
Because of this very slow decay the automatic windowing algorithm leads to
signi cant underestimates of even when the window factor c is as large as
int A
e
or We therefore de ned a mo di ed estimator by extrap olating t
int A AA
b
e b
prop ortionally to t in the region b eyond the window i e M M t for
int A
AA
t M and cutting o the sum at a much larger time M of order
exp
M
X
b b
e b b
t M M log M M C
int A
AA AA
t= M
Since M M we approximated the second sum by an integral Here M is
de ned as b efore by the automatic windowing algorithm In the absence of any
e
precise knowledge of we to ok M N f luckily is not very sensitive to
exp int A
b
b
the choice of M b ecause of the logarithmic dep endence and b ecause M
AA
In Table we show the results for a typical run d N for the
2 2 2
observables A R R R T as a function of the window factor c
K ar p Luby R=150
e g m
Time is measured in units of pivots Note that
b
b
a The standard windowing estimates are a factor smaller than the
int A
e
mo di ed estimates even at c
int A
b
b
are slowly but relentlessly increasing as a function of c b The estimates
int A
it is p erfectly plausible that they will double or triple by the time c reaches
in nity
e
c The estimates are roughly stable as a function of c as so on as c
int A
However they do show some uctuation b ecause they are based on extrap o
b
b
lation from a single noisy p oint M
AA
e
We therefore chose as our nal estimate the average of for the eleven values
int A
c by taking the average we reduce the uctuations mentioned in
c
This whole pro cedure is of course inelegant and ad hoc But it do es work
reasonably well we exp ect that the estimates of are accurate to ab out
int A
This is not go o d enough for a serious study of dynamic critical b ehavior but it is
go o d enough for our present purp ose which is merely to set error bars on the static
quantities A In the future we hop e to devise b etter metho ds for analyzing time
series with slow decay of the auto correlation function
D Remarks on the Field Theoretic Estimates of
Universal Amplitude Ratios
The critical exp onents and universal amplitude ratios asso ciated with p olymers
in a go o d solvent can b e extracted from any family of theories which intersects
transversally the domain of attraction stable manifold of the go o d solvent xed
p oint H One such family which has no sp ecial status at H but is computa
GS GS
tionally convenient is the continuum Edwards mo del or what is equivalent the
4
n continuum eld theory This mo del b ecomes critical i e crosses the
when its bare coupling constant self avoidance parameter stable manifold of H
GS
z tends to
There are two main approaches to the quantitative study of the continuum
Edwards mo del in the limit z
Perturbation expansion in the coupling constant z at xed dimension d
or d In this case the problem is to estimate the asymptotic b ehavior
as z from the rst few terms of a p erturbation expansion around z
Expansion in d Here the critical exp onents and limiting amplitude
ratios which corresp ond to the limit z can b e obtained directly from a
suitable renormalization group analysis The problem is then to extrap olate
to or
In this app endix we want to summarize the results for the universal amplitude ratios
2 2
hR i hR i and which have b een obtained by each of these metho ds
g e R
and to make some comments on their extrap olation
Perturbation expansion at xed dimension d Let z z and h z
R S
b e the conventional expansion and second virial factors of the continuum Edwards
mo del One then has
2 2
z z z D
S R
d
z z h z z D
S
d
z z h z z D
R
R
The known p erturbation series in d are as follows
2
z z z
3 4 5
z z O z D
p
2 3 4
z z O z D z z
p
2 3 4
z z O z D z z
R
Crude estimates of can b e obtained from the and Pad e
approximants
554 3395
z
105 2592
D z
[1 1]
3395 16
z
3 2592
2
z z
z D
[2 2]
2
z z
More sophisticated extrap olations have and yielding
[2 2] [1 1]
b een p erformed by Shanes and Nickel who nd Our
Monte Carlo value is
Likewise crude estimates of and can b e obtained
R
R
from the Pad eapproximants
z
p
D z
[1 1]
1769 896 2
z
105
z
p
z D
R [1 1]
254 128 2
z
15
yielding and Our Monte Carlo values are
[1 1] R [1 1]
and resp ectively
We can also try the direct renormalization approach of des Cloizeaux Conte
2
and Jannink de ne the e ective exp onent z z d log z dz
R
R
which approaches the limiting value as z re express and
R
R
as functions of and extrap olate these series to using the b est estimate
R R
R
of If we carry out this last step by the most naive metho d imaginable namely
straight evaluation of the cubic p olynomial at we
R
obtain and which is not bad at all for such a short series
R
Expansion in d The limiting universal ratios and can b e
R
44
evaluated in dimension d in p owers of the results are
2 3
O D
2
3
log O D
2
3
log O D
R
Evaluating these at yields
D
D
D
R
We remark that the Pad e approximants for all three quantities have p oles
b etween and and so are unreliable for
45
Des Cloizeaux fo otnote has suggested to augment the expansions
D D by enforcing the known exact values at d which are
1 2 1 2
see and This
R
pro duces the cubic p olynomials
2 3
D
2 3
log log D
1 2
2 3
1 2
log log D
R
Evaluating this at yields
D
D
D
R
44
The expansion for is from see also equation of The expansion for is
R
from equation of see also equations of In these works
R
is called g To establish the connection compare equations and of The
expansion for can then b e derived from these two Alternatively it can b e found as the limit
of equation a of
45
See also pp and
For this mo di cation has actually worsened the agreement with the Monte Carlo
this mo di ed value b oth in d and in d On the other hand for and
R
expansion prediction is amazingly close to the correct value b oth in d and
d It would b e useful to obtain a b etter understanding of whether this is a
3
coincidence or not p erhaps by calculating the O term in
Acknowledgments
We wish to thank Jim Barrett Sergio Caracciolo Bertrand Duplantier Michael
Fisher Peter Grassb erger Tony Guttmann Bernie Nickel George O Brien Enzo
Orlandini Andrea Pelissetto Carla Tesi and Stu Whittington for many helpful
conversations and corresp ondence The computations rep orted here were carried out
on various IBM RS and H Sparcstation Silicon Graphics Crimson
and Convex C machines at New York University
The authors research was supp orted in part by an op erating grant from the
Natural Sciences and Engineering Research Council of Canada N M U S De
partment of Energy contract DE FG ER A D S U S National Sci
ence Foundation grants DMS and DMS A D S and by a New
York University Research Challenge Fund grant A D S Acknowledgment is also
made to the donors of the Petroleum Research Fund administered by the Ameri
can Chemical So ciety for partial supp ort of this research under grants AC
and AC B C A D S During part of the p erio d of research B L was a
American Chemical So ciety Petroleum Research Fund Fellow
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J Dayantis and J F Palierne J Chem Phys
L A Johnson A Monge and R A Friesner J Chem Phys
F Shanes and B G Nickel Calculation of the radius of gyration for a linear
exible p olymer chain with excluded volume interaction J Chem Phys to
app ear
J Dayantis and J F Palierne Phys Rev B
A J Guttmann J Phys A
B G Nickel Physica A
K E Newman and E K Riedel Phys Rev B
A D Sokal Fundamental problems in the static scaling b ehavior of high
molecular weight p olymers in dilute solution I I Critical review of the exp er
imental literature in preparation
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Phys Ed
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H Utiyama S Utsumi Y Tsunashima and M Kurata Macromolecules
B App elt and G Meyerho Macromolecules
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B Chu R Xu T Maeda and H S Dhadwal Rev Sci Instrum
K B Strawbridge F R Hallett and J Watton Can J Appl Sp ectroscopy
A D Sokal Optimal statistical analysis of static light scattering data from
dilute p olymer solutions in preparation
M Benhamou and G Mahoux J Physique Lett L
C Domb and F T Hio e J Chem Phys
M van Pro oyen and B G Nickel The second virial co e cient for self avoiding
walks on a lattice in preparation
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NY
H Yamakawa Modern Theory of Polymer Solutions Harp er and Row New
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2 4 4 4 4
N hT i hT i var T hU i hV i hV i
B ar r ett K ar p Luby
Table Quantities relevant to the Barrett and Karp Luby algorithms as a function
of N N N for dimensional self avoiding walks Error bars are one standard
1 2
deviation
2 4 4 4 4
N hT i hT i var T hU i hV i hV i
B ar r ett K ar p Luby
Table Quantities relevant to the Barrett and Karp Luby algorithms as a function
of N N N for dimensional self avoiding walks Error bars are one standard
1 2
deviation
2 2 2
N hR i hR i hR i hT i f
e g m
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
Table The results of our runs in dimension d Errors are one standard
a
deviation indicates a p ossible minor bug in measurement see text
2 2 2 2
N hR i hR i hR i hR i
g e m e
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
Table Universal amplitude ratios in dimension d Errors are one stan
a
dard deviation based on triangle inequality indicates a p ossible minor bug in
measurement see text
2 2
N hR hR hT i f i i
e g
a
a
a
a
a
a
a
a
a
a
a
a
a
a
Table The results of our runs in dimension d Errors are one standard
a
deviation indicates a p ossible minor bug in measurement see text
2 2
N hR i hR i
g e
a
a
a
a
a
a
a
a
a
a
a
a
a
a
Table Universal amplitude ratios in dimension d Errors are one stan
a
dard deviation based on triangle inequality indicates a p ossible minor bug in
measurement see text
2
N a b
min
DF level
DF level
DF level
DF level
DF level
2 2
Table Fits hR i hR i a b N with a b all variable for
g e
dimensional SAWs The true error bars are probably of those indicated
here see text
2
N a b
min
DF level
DF level
DF level
DF level
DF level
DF level
Table Fits a b N with a b all variable for dimensional
SAWs The true error bars are probably of those indicated here see text
d d d d
har d spher e
N =1
N =2
N =3
N =4
N =5
N =6
N =7
har d spher e
Table for short chains rounded to four decimal places from exact enumer
N
ations along with our Monte Carlo values for
2 2 2
A R A R A R A T
K L R=150
e g m
b b b b
e e e e b b b b
c 2 2 2 2 2 2
int T int T
int R int R int R int R int R int R
e g m e g m
b
b e
and mo di ed estimate for the Table Standard windowing estimate
int A int A
2 2 2
T as a function of window factor c for R R observables A R
K ar p Luby R=150
m g e
SAWs on the square lattice at N Time is measured in units of pivots
2 2
Figure Estimated exp onent from pure p ower law ts to hR i 2 hR i 3
e g
2
and hR i and estimated exp onent from pure p ower law t to hT i
m 4
plotted versus N Note the go o d agreement with the b elieved exact value
min
and with the hyperscaling relation d as so on as N
4 min
Figure Estimated exp onent p from pure p ower law t to pivot algorithm accep
tance fraction f plotted versus N
min
Figure Interpenetration ratio versus N for dimensional SAWs Note that
and is constant within error bars for N varies little for N
2 2
Figure Estimated exp onent from pure p ower law ts to hR i 2 and hR i
e g
3 and estimated exp onent from pure p ower law t to hT i plotted
4
versus N Note the very strong corrections to scaling which lead to erroneous
min
4
exp onent estimates unless one takes N
min
Figure Estimated exp onent p from pure p ower law t to pivot algorithm accep
tance fraction f plotted versus N
min
2 2
Figure Estimated exp onent from pure p ower law ts to hR i 2 and hR i
e g
3 and estimated exp onent from pure p ower law t to hT i plotted
4
0 5
Note the very roughly linear b ehavior in agreement with the b elief versus N
min
that
1
pow er pow er
Figure Pairs for which the t to the Ansatz AN BN pro
2
duces a acceptable at the signi cance level i e one standard deviation
2
3 and hT i Observables are hR i
g
Figure Interpenetration ratio versus N for dimensional SAWs Note that
is a decreasing and convex function of N in agrant disagreement with the
prediction of the two parameter renormalization group theory
0 56
Figure Interpenetration ratio versus N for dimensional SAWs The
0 56
regression line is N Note the excellent linearity of the
plot
P
Q
u
u
C
M
s
H
1
u
H
2
j
u
U e
u
H
3
H
u
1
e
q
u H
2
R
e
u
z
H
u
u
3
H
GS
M
u
u
H
Figure Wilson de Gennes type renormalization group ow on the critical sur
face H resp H is the Gaussian resp go o d solvent xed p oint M resp
s
GS
M is the stable resp unstable manifold of H Case Ia Mo dels in the go o d
u
solvent regime may have correction to scaling amplitudes that are either negative
P R or p ositive Q Case I I The initial Hamiltonians H approach the stable
n
ef f n
manifold while the low energy e ective Hamiltonians H R H approach the
n
n
unstable manifold
2 5
iterations for pivot Figure Averages of a R and b T over bins of width
e
algorithm on simple cubic lattice at N with parallel ro d start
2
b
b
Figure Log log plot of the sample auto correlation function t for a A R
AA
e
2
and c A T for pivot algorithm on square lattice at b A R
K ar p Luby R=20
g
N