The SwendsenWang pro cess

do es not always mix rapidly



Vivek K Gore and Mark R Jerrum

Department of Computer Science

University of Edinburgh

st Octob er

Abstract

The SwendsenWang pro cess provides one p ossible dynamics for the Q

state Potts mo del in statistical physics Computer simulations of this

pro cess are widely used to estimate the exp ectations of various observ

ables random variables of a Potts system in the equilibrium or Gibbs

distribution The legitimacy of such simulations dep ends on the rate of

convergence of the pro cess to equilibrium often known as the mixing rate

Empirical observations suggest that the SwendsenWang pro cess mixes rap

idly in many instances of practical interest In spite of this we show that

there are o ccasions on which the SwendsenWang pro cess requires exp o

nential time in the size of the system to approach equilibrium



Address for corresp ondence Department of Computer Science University of Edinburgh

The Kings Buildings Edinburgh EH JZ United Kingdom

Introduction

The Potts mo del is a natural generalisation of the Ising mo del to an arbitrary

number Q of states or spins Congurations in the Potts mo del can b e

viewed as Qcolourings not in general prop er of the vertices of an undirected

nvertex graph One is interested in sampling congurations from a certain distri

bution known as the Gibbs distribution with the aim of obtaining estimates for

certain random variables on congurations For the ferromagnetic Potts mo del

the fo cus of this article the Gibbs distribution assigns greater probability

to congurations in which a larger number of pairs of adjacent spins are alike A

precise denition of the Potts mo del is provided in Section

In the absence of eective direct metho ds the usual approach to sampling

congurations is via the Markov chain Monte Carlo metho d The idea is

to provide the mo del with a dynamics by dening an ergo dic random walk on

congurations whose stationary distribution is the required Gibbs distribution

Provided the walk is rapidly mixing ie converges rapidly to equilibrium con

gurations may b e eciently sampled by simulating the walk for a sucient but

not excessive number of steps

A number of dierent dynamics are p ossible The simplest is to move b etween

congurations by changing one spin at a time with transition probabilities de

termined by the Metrop olis rule It is fairly easy to demonstrate situations

in which this random walk takes exp onential time in n the size of the graph to

approach equilibrium even in the ferromagnetic case A more complicated dy

namics which allows many spins to change in one step was prop osed by Swendsen

and Wang and is now widely used in computer simulations

The SwendsenWang pro cess as we shall call it app ears to converge rapidly

to equilibrium in many instances of practical interest This empirical observation

might encourage us to attempt to prove that the mixing time of the pro cess

grows not to o quickly as a function of n sp ecically that it is b ounded by a

xed p olynomial in n indep endent of the other parameters of the system Such

a result would establish the existence of an ecient approximation algorithm

more precisely a fully p olynomial randomised approximation scheme or

fpras for computing the partition function of a Qstate ferromagnetic

Potts system Such an algorithm is only known to exist in the case Q

Our main result see Prop osition for a precise statement demonstrates that

this is a vain hop e For a certain particularly simple family of Potts systems based

on the complete graph K on n vertices the socalled CurieWeiss mo del the

n

SwendsenWang pro cess is still far from equilibrium after exp onentially many

steps This counterexample is valid for all Q and for a suitably chosen

coupling constant It is an op en question whether rapid mixing obtains when

1

The antiferromagnetic mo del in which adjacent spins tend to b e unlike includes graph

colouring as a limit so rapid convergence cannot b e exp ected for any reasonable dynamics

Q the Ising mo del or if the negative result can b e extended to more

physically realistic instances of the Potts mo del for example or dimensional

lattices

The Potts mo del

The Potts mo del was introduced by R B Potts in has b een a fo

cus of much attention in the physics and mathematics communities ever since

Rather than present a detailed historical account of the mo del here we refer the

interested reader to Baxter Chap

The problem is easily stated Consider a collection of sites f ng de

noted by n each pair i j of which has an asso ciated interaction energy which

for simplicity we assume takes on one of just two values either or J In

most cases of physical interest the set E of unordered pairs of sites with non

zero interaction energy forms a regular lattice graph n E A conguration

is an assignment of spins to sites where denotes the spin at

n i

site i The number of spins is denoted by Q where Q individual spins may

simply b e denoted by the numbers in Q The energy of a conguration is

given by the Hamiltonian

X

H J

i j

ij E

where is the Kronecker function which is if its arguments are equal and

otherwise

The central problem is to compute the partition function

X

Z exp H



where is what is called the inverse temperature to b e precise k T

where T is the temp erature and k is Boltzmanns constant and the sum is over

all p ossible congurations Many of the physical prop erties of the system

can b e computed from the knowledge of Z Essentially Z is the normalising

factor in the calculation of probabilities according to the fundamental theory of

statistical mechanics the probability that the system in equilibrium is found in

state the steady state probability is Z exp H Moreover

certain logarithmic derivatives of Z corresp ond to quantities such as mean energy

and mean magnetic moment Singularities in these derivatives in the limit as

n generally corresp ond to phase transitions when a small change in a

parameter has an observable eect on the macroscopic prop erties of the system

If a small change in temp erature causes a phase transition then that temp erature

is called the critical temperature

Consider the eect of the parameter J in the Hamiltonian The high probab

ility congurations are those for which H is low Let d denote the number

of edges in E that connect sites with dierent spins in the conguration It is

easy to see that H J d and that

X

Z exp J d



We let K J K is usually known as the coupling constant in the statistical

physics community By the denition of H if J or K since is

p ositive then congurations in which neighbouring spins spins asso ciated with

a pair of sites with nonzero interaction energy are the same are preferred this

is the ferromagnetic attractive case On the other hand if J is negative the

neighbouring spins will tend to b e dierent and this is the antiferromagnetic

repulsive case

The search for ecient computational solutions to these problems has proved

extremely hard and has generated a vast b o dy of literature The reader is re

ferred to Chapter of Baxter where some sp ecial cases have b een considered

A huge amount of computational eort has also b een p oured into numerical solu

tions esp ecially for regular lattices

Although this problem arises in statistical physics it has a very interesting

connection with theoretical computer science It is another example of a signi

cant combinatorial enumeration problem which is Pcomplete and is hence

apparently intractable in exact form This is an intriguing class of problems

and includes the problems of computing the volume of a convex b o dy and the

p ermanent of a matrix The Potts mo del also turns out to b e one of the many

sp ecialisations of the famous Tutte p olynomial in graph theory The reader is

referred to Welsh p for more on this interesting connection

A lot of research eort has b een devoted to nding ecient approximation

algorithms for Pcomplete problems where by ecient we mean that the al

gorithm is guaranteed to run in time p olynomial in the number of sites n Ran

domness has played a ma jor role in this area and ecient randomised approx

imation algorithms have b een given for computing the volume of a convex b o dy

and estimating the p ermanent of a dense matrix as well as for many other

problems Each of these algorithms is a fully p olynomial randomised approx

imation scheme fpras ie one that pro duces solutions which with very high

probability fall within arbitrarily small error b ounds sp ecied by the user the

price of greater accuracy b eing a mo dest increase in runtime

Most of these algorithms use Markov chain simulation This approach has

b een used extensively over the years in the eld of physics and in the last ten

years or so it has b een used by researchers in computer science to provide fully

p olynomial randomised approximation schemes for many problems For further

information on this approach and its applications refer to the surveys by Kan

nan and Jerrum and Sinclair

Single spin ip pro cess

It is in fact very easy to use the Markov chain approach to approximate the

partition function of a Potts system The state space is simply the set of

all p ossible congurations Let denote the steady state probability of the

conguration as describ ed earlier Transition probabilities from the current

state are mo delled by the following pro cedure

choose a site i n and a spin s Q uniformly at random

assign spin s to site i to get a new conguration and let the probability

of accepting the move p minf g

acc

with probability p let the next state b e and with probability p

acc acc

let the next state b e itself

The pro cedure describ ed ab ove is an example of what is known in the stat

istical physics community as the Glaub er dynamics or the single spin ip

dynamics of the Potts mo del The acceptance condition used here is called the

Metropolis rule which is familiar in the computer simulation of mo dels in statist

ical physics Unfortunately the Markov chain describ ed ab ove has not b een

proved to b e rapidly mixing In fact when Q ie when we have an Ising

system it can b e shown that the chain is not rapidly mixing in the ferromagnetic

case when K is suciently large It is well known that ferromagnetic Ising sys

tems typically exhibit a phase transition at a certain value of the parameter

for values of ab ove this critical value the system settles into a state in which

there is a prep onderance of one or the other of the two spins

A signicant theoretical advance came in when Jerrum and Sinclair

describ ed the rst provably ecient approximation algorithm for the partition

function of an arbitrary ferromagnetic Ising system They used a completely

dierent Markov chain one in which the state space consists of all the subgraphs

of the interaction graph G They proved that this Markov chain is rapidly mixing

at all temp eratures and this fact can b e used to estimate the partition function

of the system Unfortunately their approach do esnt seem to generalise to Q

The antiferromagnetic case seems to b e even harder in fact for the Ising

mo del Jerrum and Sinclair following Barahona proved that the existence

of even an fpras is highly unlikely Similar results for the Potts mo del have b een

proved by Welsh p However it is worth mentioning that if Q

where is the maximum degree of a vertex in the interaction graph G then the

Markov chain describ ed ab ove can b e shown to b e rapidly mixing and yields an

fpras This was shown for the zero temp erature case by Jerrum and

then extended to arbitrary temp erature at least for Q by Sokal The

latter result follows directly from the standard pro of of the Dobrushin uniqueness

theorem combined with Salas and Sokals verication of the hypotheses

of that theorem for the case of the antiferromagnetic Potts mo del with Q

The random cluster mo del

Before introducing the SwendsenWang approach for approximating the partition

function for the Qstate Potts mo del it would b e instructive to lo ok at a related

mo del called the Random Cluster mo del which was introduced by Fortuin and

Kasteleyn in We have the interaction graph G n E just as b efore

however there are no spins on the sites The b ehaviour of the system dep ends

on the formation of bonds b etween pairs of interacting sites and the clusters

connected comp onents in graph terminology formed by these b onds There are

two parameters asso ciated with the mo del a probability p of the formation of

a b ond b etween two interacting sites and a weighting factor Q When this Q

is a p ositive integer it corresp onds to the Q in the Potts mo del however in the

random cluster mo del Q may b e an arbitrary nonnegative real number The

partition function for this mo del is given by

X

jAj jE jjAj C A

Z p p Q

AE

where A denotes the set of interactions that form a b ond and C A is the number

of clusters connected comp onents in the b ond graph A The sum is over all

p ossible subgraphs of G

It turns out that the Qstate Potts mo del is equivalent to the random cluster

mo del with p exp K where K is the coupling constant as describ ed

earlier and the parameter Q is common to b oth mo dels The equivalence is

close even at the microstate level to obtain a Potts conguration from a random

cluster conguration simply assign a spin from Q indep endently and uar to

each cluster Note that the random cluster mo del eectively generalises the Potts

mo del to an arbitrary p ossibly nonintegral p ositive number of spins

The SwendsenWang approach

This approach is based on the b ond graph and cluster view of the Potts mo del

describ ed in the previous section Unlike the single spin ip dynamics this ap

proach is not lo cal in that a single transition can aect a large number of sites

Each transition can b e describ ed as a twostep pro cess and the connection with

the random cluster mo del is very clear Let the current Potts conguration b e

denoted by The next conguration is obtained as follows

Let A E b e the subset of edges that form a b ond ie ones with the same

spin on b oth incident sites Each of the edges in A is retained indep endently

with a probability p exp K this gives a subset A of A

Using A as the set of edges connected comp onents clusters are formed

For each cluster a spin is chosen uniformly at random from Q and all

sites within the cluster are assigned that spin

That the Markov chain with transitions dened by this exp eriment is ergo dic is

immediate that it has the correct distribution is not to o dicult to show See

for example Edwards and Sokal

This Markov chain seems to work very well in practice however it has not

b een proved to b e rapidly mixing for any class of graphs The nonlo cal nature

of the transitions seems to allow the chain to move more freely within the state

space thus avoiding the p ossible constrictions that might result at low temp erat

ures Our result deals with the case where the interaction graph is the complete

graph K and we prove that the chain is not rapidly mixing in this case for

n

Q

A rstorder phase transition

The slow mixing rate of the SwendsenWang pro cess is connected with what is

known as a rstorder phase transition which we now investigate in the context

of the Potts mo del on the complete graph CurieWeiss mo del We exhibit two

distinct kinds of congurations that account for all but an exp onentially small

fraction of the partition function Z In fact by tuning the coupling constant

we arrange that the two kinds of congurations make a roughly equal contribu

tion to Z Such a system is said to b e in a mixed phase and the two kinds of

congurations are called coexisting phases This value of the coupling con

stant is usually referred to as its critical value denoted here by K The phase

cr

transition reects a crucial instability in the mo del in the following sense When

K K the system prefers the so called ordered phase one of the spins domin

cr

ates As K is decreased ie temp erature is increased the system go es into a

mixed phase at K K and then makes an abrupt transition to the disordered

cr

phase each of the Q spins app ears roughly the same number of times when

K K

cr

We consider the case where K K and in Section we show that the

cr

SwendsenWang pro cess only very infrequently makes a transition b etween the

co existing phases which results in a slow mixing rate

Consider an arbitrary conguration of the Qstate Potts system Recall

that the equilibrium probability of is given by

Z exp K d

where d is the number of pairs of sites in E with dierent spins We choose

K cn where c Q is a constant dep ending on Q See for an

explicit expression for c For the complete graph K since all interactions are

n

present the only relevant observable quantities are the number of dierent spins

and the sizes of these spin classes Let n n n b e the vector whose ith

Q

comp onent is the size of the ith spin class of we say that n is the type of

Note that

Q

X

n n d

i

i

Since is a function only of type we may write n to denote the value of

for any conguration of type n

In equilibrium the probability of b eing in a conguration of type n is N n

n where

n

N n

n n n

Q

denotes the number of congurations of type n Using Stirlings approximation

Q

X

Q

N n n exp a ln a n a

i i

i

where a a a nn and a is an error term in general jaj

n

O log n but the tighter estimate jaj O holds if it is known that a

for some constant The implicit constants dep end of Q and

only

From recalling K cn we have

Q

X

c

n Z exp a n

i

i

Therefore

Q

Pr has type n Z n exp ff an ag

where

Q

X

c

g a f a

i

i

cx x ln x and g x

In order to identify the congurations that have the largest weights we need

P

Q

to maximise f which in turn means that we need to maximise g a in

i

i

P

Q

the region dened by a for all i and a This is clearly a closed

i i

i

region viewed as a set in Q dimensional Euclidean space and we use R

to denote it We now pro ceed to lo ok at the b ehaviour of g in the interval

The following are easy observations

if we dene g then g is continuous in

g x cx ln x is dened in and tends to as x

g x has a unique minimum in at x c such that g c ln c

g x in c and g x in c

Prop osition Let a a a be a local maximum point of f Then a

Q

satises the fol lowing properties

i a lies in the interior of R

ii Either a Q for al l i or there are and such that c

i

and a f g for al l i

i

iii If a is such that the a are not al l equal then there is a unique component a

i j

such that a the other components a with i j satisfy a

j i i

Furthermore g g

Pro of

i Supp ose on the contrary that a is such that a and a Since

i j

g x as x a and g a is nite we can increase f by setting

j

i

a and a a where is suciently small

i j j

ii At any lo cal maximum it must b e the case that g a g a for all i

i j

and j For supp ose g a g a for some i j Then a small p erturba

i j

tion of to a and a either a a and a a or the other way

i j i i j j

round dep ending on the values of g a and g a would cause f a to

i j

increase Since g x is unimo dal in a f g for all i where and

i

are on either side of the minimum of g

iii Supp ose on the contrary that a a where j k Since g

j k

setting a a and a a would cause f a to increase

j j k k

If we now set

Q lnQ

c

Q

it is routine to verify that the following three choices for a satisfy prop erties

iiii in the statement of Prop osition

S a Q for all i Q

i

S a QQ for all i Q and a Q Q

i Q

S a Q for all i Q and a

i Q

2

Beyond a certain value c of c the function f can b e shown to have two lo cal maximum

0

p oints and a lo cal minimum p oint Our chosen value of c is the unique value at which the two

lo cal maxima b ecome equal to yield a global maximum Much of the analysis has b een done in

alb eit in a somewhat dierent context

We shall shortly see that they are the only solutions satisfying prop erties i

iii Note that the ordering of subscripts on the spin classes is not signicant as

explained earlier

Claim The rst two solutions SS above are the only local maximum

points of f in R and they both correspond to the global maximum of f The

nal solution S is a local minimum

Pro of It is clear from Prop osition that any maximum p oint of f should have

the form a for all i Q and a Q for some

i Q

in Q satisfying

h g g Q

Now h cQ c ln Q ln and

Q

cQ h cQ

Q Q

Setting h we get the quadratic equation Q cQ which

implies that h has at most two turning p oints and hence at most three zeros

in the interval Q Since QQ Q and Q

all satisfy h they are in fact the only solutions to that equation We

conclude that SS are the only choices for a consistent with the conditions

of Prop osition and hence they must cover all the lo cal maximum p oints of

f a

We now pro ceed to show that solutions S and S corresp ond to the global

maximum of f and that S do es not In fact it is a lo cal minimum p oint

Since by Prop osition we are only interested in solutions of the form a for

i

all i Q and a Q we may view f as a function of

Q

the single variable Accordingly dene f f For the value of

c chosen ab ove we have by direct calculation

Q lnQ

f Q f QQ ln Q

QQ

and

Q lnQ

f Q ln

Q

Denote by dQ the dierence b etween and

Q lnQ

dQ f Q f Q ln Q ln

Q

Then dQ for all Q To verify this observe that for Q

Q ln Q

dQ ln Q ln ln Q ln ln Q

Q

The cases Q can b e checked separately Thus S and S are global

maximum p oints of f and are the only such

Note that dQ f Q f Q is very small for smaller values

of Q eg of the order of for Q however since f app ears in the

exp onent and is multiplied by n the actual dierence in weights of these two

types of congurations is quite large even for fairly small values of n

Denote by B the set of all p oints in R that are within Euclidean dis

tance of the balanced maximum p oint Q Q and by B the p oints

that are within distance of any one of the other maximum p oints B is

a Q dimensional ball and B a union of Q such balls Let

resp ectively b e the set of congurations whose type n lies in nB re

sp ectively nB The following result summarises what we have discovered

Prop osition For any

Q

i Pr n

Q

ii Pr n and

n

iii Pr e

The implicit constants depend only on Q and

Pro of Let a R b e chosen so that every comp onent of a has value either

in or i n for some integer i and let n na observe that all con

gurations of type n are in provided n is suciently large Then

jf a f Q Q j O n and hence by n comes within a con

stant factor of maximising Pr is of type n over all types n Note that we are

op erating within the jaj O regime Since the total number of distinct

Q

types is O n we have part i Part ii is proved in a similar manner

Finally note that the supremum of f a over the region a R n

is strictly less than the supremum over the whole of R Part iii follows by

combining this observation with

Dynamics

It is clear from Prop osition that the single spin ip pro cess describ ed in Sec

tion will converge only very slowly to equilibrium since it is dicult to escap e

from either of the neighbourho o ds or using small steps However

the SwendsenWang pro cess is able to change large blo cks of spins in one step

which at rst sight seems to give it a signicant advantage Our main result

Prop osition b elow suggests that this advantage may on o ccasion b e illusory

Before we present a formal pro of it would b e useful get an intuitive feel as

to why we exp ect Prop osition to b e true Let denote the current Potts

conguration Note that the SwendsenWang pro cess only considers edges that

form a b ond so that the conguration may b e viewed as a collection of smaller

complete graphs one for each spin in Q Let n n n denote the sizes

Q

of these graphs Let G denote the standard random graph mo del in which an

p

undirected vertex graph is formed by adding indep endently for each unordered

pair of vertices u v an edge connecting u and v with probability p Step of the

pro cess essentially creates Q random graphs G one of each size n i Q

n p i

i

where the probability of retaining an edge is p exp K Recall that

K cn where c Q so that for large n p cn For a carefully chosen

value of c we exp ect two p ossibilities

Just prior to Step if all the spin classes in have roughly the same size

nQ then for any such class the probability p of retaining an edge

can b e written as d where d cQ so that d A wellknown result

in the theory of random graphs tells us that with very high probability

the spin class will break up into very small comp onents of size O log

so that at the end of Step after assigning random spin values to these

very small comp onents with high probability we again end up with all spin

classes having roughly equal size

Just prior to Step if there is one very large spin class in and all the

other spin classes are very small then the value of d as ab ove would b e

for the large class whereas it would b e for the other classes We can

now app eal to results ab out the giant comp onent in random graphs

to say that with high probability at the end of Step there would b e

one large comp onent and all the other comp onents would b e very small

This means that at the end of Step we exp ect with high probability a

conguration similar to the one b efore Step The choice of c determines

how close the new conguration would b e to the previous one

Prop osition shows that for a carefully chosen value of c dep ending on Q

a Qstate Potts system tends to settle in one of the two kinds of congurations

mentioned ab ove and the probability of making a transition from one kind to

the other is very small Our pro of utilises some standard b ounds on the tails

of distributions of sums of indep endent rvs that we state here for convenient

reference

Lemma Cherno Let the random variable Z have distribution Bin p

where Bin p is the binomial distribution with parameters and p Then for

any real

p

e

Pr Z p

Pro of See Theorem

Lemma Hoeding Let Z Z be independent rvs with a Z b

k i i i

P

k

b

Z Then for for suitable constants a b and al l i k Also let Z

i i i

i

any

k

X

b b

Pr jZ Exp Z j n exp n b a

i i

i

Pro of See Theorem

Let G denote the standard random graph mo del as b efore Supp ose that

p

p d with d a constant and G is selected according to the mo del G It

p

is a classical result that with probability tending to as the connected

comp onents of G all have size O log We require a fairly crude large deviation

version of this result

Lemma Let G be selected according to the model G where p d and

p

d is a constant Then the probability that G contains a component of

p p

size exceeding is exp

Pro of Following Karp we consider a simple sto chastic pro cedure for grow

ing a connected comp onent of G from sp ecied vertex s Let D fsg and

P At step t D will b e the set of discovered vertices those that have

t

b een shown to b e connected to s and P D the set of pro cessed vertices

t t

If D P we are done D is the connected comp onent of G containing s Oth

t t t

erwise we select v D n P and let D D Gv and P P fv g

t t t t t t

where Gv denotes the set of neighbours of v in G Note that the termination

condition is equivalent to jD j t

t

p

steps with very high We must show that this pro cess terminates within

probability We do this by comparing the evolution of D against another se

i

quence of random variables X dened by X and X X Bin d

i t t

where Bin p is the binomial distribution with parameters and p If Y

and Z are random variables taking nonnegative integer values then Y is said

to stochastically dominate Z if PrY i PrZ i for all i It is an

elementary fact that sto chastic domination is preserved by addition Now it is

easily checked that X X sto chastically dominates jD j jD j and so X

t t t t t

sto chastically dominates jD j But X is clearly distributed as Bint d

t t

so we can estimate the probability that X is large using a Cherno b ound In

t

particular letting t td in Lemma the probability that X exceeds t

t

is b ounded as follows

PrX t Pr X t

t t

Pr X td

t

Pr X Exp X

t t

td

e

p

Setting t b c and noting that d as we obtain

p

PrjD j t PrX t exp

t t

where we have used the fact that X sto chastically dominates jD j But jD j t is

t t t

the event that the comp onent building pro cedure has not terminated at or b efore

time t ie that the connected comp onent of G containing s has size greater

p

than t b c Multiplying by we obtain a b ound on the probability that any

p

connected comp onent in G has size exceeding this small extra factor may b e

absorb ed by the notation

Prop osition Suppose Q is an integer c Q Q lnQ

and consider a Potts system on K with coupling constant K cn Let

n

be suciently smal l and let and be as in Proposition Starting

at any conguration the expected time T for the SwendsenWang

p

n process to reach a conguration T is exp

Pro of Supp ose the conguration t at time t is an arbitrary member of

By denition of the size of any spinclass of is b ounded ab ove by

Q n Focussing attention on a particular spinclass of size the

set A constructed in step of the SwendsenWang pro cess is the edge set of

a complete graph K on vertices and the set A is the edge set of a random

graph G selected according to the mo del G where

p

c c

K

p e

n Q

Since cQ we have p d where d provided is suciently small

p

By Lemma with probability exp all connected comp onents of G

p

have size at most Since Q n n the same statement holds

with n replacing Similar arguments apply to the other spinclasses so with

p

probability exp n all the connected comp onents formed in step

p

of the SwendsenWang pro cess have size at most n

Let b e the number of such comp onents and s s b e their resp ective

sizes The exp ected size of a spinclass constructed in step of the Swendsen

p

n Wang pro cess is nQ and b ecause there are many comp onents at least

we exp ect the actual size of each spinclass to b e close to the exp ectation We

quantify this intuition by app ealing to a Ho eding b ound Fix a spin and

b

dene the random variables Y Y and Y by

s if the ith comp onent receives spin in step

i

Y

i

otherwise

P

b b

and Y Y Then Exp Y nQ and by Lemma for any

i

i

X

b b

s Pr jY Exp Y j n exp n

i

i

p

n exp

since

X X

p

s n n s

i

i

i i

p

Q we see Similar b ounds apply of course to the other spins Cho osing

p

that with probability exp n the size of every spinclass in t

p p

lies in the range Q Q n Q Q n but this condition implies

t The claimed result follows easily

Note that Prop osition only proves that starting from any conguration in

the SwendsenWang pro cess requires exp onential time to reach a cong

uration in However since the SwendsenWang pro cess is reversible it is

clear that the same statement also holds if we interchange the roles of and

in the Prop osition

Acknowledgements

We thank Alan Frieze and Boris Pittel for invaluable p ointers to relevant literat

ure Peter Winkler for an enlightening discussion on Gibbs measures and phase

transitions and Alan Sokal for his careful reading and constructive criticism of

an early version of this pap er

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