Monte Carlo Studies of Self-Avoiding Walks and Loops

Monte Carlo studies of Self-Avoiding Walks and Loops

Tom Kennedy

Department of Mathematics, University of Arizona Supported by NSF grant DMS-0501168 http://www.math.arizona.edu/etgk

Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.1/41 ntreal – p.2/41 lready Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo

Preface The organizers wrote “We would like to encouragedevelopments you rather to than speak to on delivergiven ongoing a on or lecture several recent that occasions.” you have a I have not given this talk before. I will not give this talk again. ntreal – p.2/41 gs I do lready Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo

Preface The organizers wrote “We would like to encouragedevelopments you rather to than speak to on delivergiven ongoing a on or lecture several recent that occasions.” you have a I have not given this talk before. I will not give this talk again. I will mainly talk aboutnot ongoing understand. work with an emphasis on thin ntreal – p.3/41 3 / 8 Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo interior point vs. endpoint SAW vs. SLE comparison with Werner’s measure on self-avoiding loops

Outline 0. Review: Def of SAW, conjectured relation1. to Bond SLE avoiding random walk 2. Two saw’s - comparison with Cardy/Gamsa3. formula Distribution of points on SAW and SLE 4. Bi-inﬁnite SAW as a self-avoiding loop • • • • • ntreal – p.4/41 ce. Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo . Then let lattice spacing go to zero. step, nearest neighbor walks in the upper half plane, N Deﬁnition of SAW →∞ N Take all starting at the origin which doGive not them visit the any uniform site probability more measure. Let than on ntreal – p.5/41 ’s 3 / 8 r of steps Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo 3 / 8 . N β

Relation to SLE All simulations supporting the conjecture have been for SAW with a weight Simulations of SAW support the conjecture SAW in other geometries typicallyN requires a variable numbe Do this in the upper half plane Conjecture (LSW) : the scaling limit of the SAW is chordal SLE Note: with a ﬁxed number of steps. ntreal – p.6/41 al SAW. 1800 that self-avoid in the sense N suppressed. 1780 Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo 1760 entropically 1740 1720 1700 1220 1200 1180 1160 1140 1120

1. Bond-avoiding walk Take all nearest neighbor walksthat of they length do not traverse the sameLarge bond loops more are than allowed, once. but Everyone expects this model to have same scaling limit as usu ntreal – p.7/41 s "1" "2" "3" "4" Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo

Bond and site avoiding SAW’s Picture of two bond avoiding walks and two site avoiding walk ntreal – p.8/41 900 850 800 Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo 750 700

Bond and site avoiding SAW’s 650 5640 5620 5600 5580 5560 5540 5520 5500 5480 5460 Blow up of previous picture showing small loops in bond SAW Answer: 1=bond, 2=bond, 3=site, 4=site ntreal – p.9/41 curve . 1% . 1 0 0.8 Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo Schramm’s formula bond SAW simulation 0.6 theta/pi 0.4 0.2

Probability of passing right 0 1 0

0.8 0.6 0.4 0.2 probability of passing right passing of probability Schramm gave an explicit formulapasses for to the the probability right the of SLE a ﬁxed point. Difference is about ntreal – p.10/41 . 3 Y Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo / 8 X

RV tests 0 Lawler, Schramm and Werner gaveprobability an of explicit certain formula for events the for SLE ntreal – p.11/41 1 0.9 0.8 0.7 0.6 Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo 0.5 Distance to 1 0.4

RV 0.3 X 0.2 0.1 Exact LSW result

Test using bond SAW simulation 0 1 0

0.8 0.6 0.4 0.2 CDF’s ntreal – p.12/41 1 0.9 0.8 0.7 Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo 0.6 0.5 Distance to 1

RV 0.4 X 0.3 0.2 0.1

Test using 0 0

-0.01 -0.02

0.005 -0.005 -0.015 -0.025 difference in CDF’s in difference ntreal – p.13/41 20 18 16 14 LSW exact result Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo bond SAW simulation 12 10 8 RV Y min height along vertical line 6 4 2

Test using 0 1 0

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 CDF’s ntreal – p.14/41 20 18 16 14 Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo 12 10 8

RV min height along vertical line Y 6 4 2

Test using 0 0

0.01 -0.01

0.005 -0.005 difference of CDF’s of difference ntreal – p.15/41 e origin Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo

2. Two SAW’s Consider two SAW’s in thewith upper the half condition plane that both they starting do at notUniform th intersect probability each measure other. ntreal – p.16/41 gin. int is left th ) ) 4 4 t t +5 +5 2 2 t t ))(1 + 6 ))(1 + 6 t t 2 2 ) ) ) 2 2 2 t t t 4 Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo 6 arctan( (1 + (1 + 5(1 + π − + 6 arctan( π π π 30 = 30 )+(3 )+(3 2 middle 2 t t P their formulae are 3 / (13 + 15 (13 + 15 t t 2 =8 2 . ) − κ θ = ( =

Two SLE’s cot right left = P P t Cardy and Gamsa considered two SLE curves starting atUsed the boundary ori CFT to deriveof the both probabilities curves, that in a thecurves. given middle For po of the curves or to the right of bo Let ntreal – p.17/41 1 0.8 exact SAW’s 0.6 Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo theta/pi 0.4 0.2

SAW simulation vs. CFT formula 0 1 0 0.8 0.6 0.4 0.2 ntreal – p.18/41 1 left right middle 0.8 0.6 Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo theta/pi 0.4 0.2

Differences - SAW’s vs exact 0 0 0.004 0.003 0.002 0.001 -0.001 -0.002 -0.003 -0.004 ntreal – p.19/41 restricted to 2] , [0 steps of a SAW with N "100K steps" "200K steps" Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo the same as SLE on is the same as the ﬁrst 1] , [0 is not steps N

3. Distribution of points on SAW and SLE . steps. 1] , N SAW with 2 SLE on time interval [0 ntreal – p.20/41 es. the steps. N . . 1 1 Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo steps N steps for a SAW with 2 N/

SAW vs SLE The endpoint of SAW with The point after SLE at time (half-plane capacity) • • • We consider the distribution ofdistance the from random the variable origin which to is various points on theWe random compare curv the following All RV’s are normalized to have mean ntreal – p.21/41 2 1.5 midpoint of SAW endpoint of SAW SLE at fixed capacity Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo 1 Distance from origin 0.5

SAW vs SLE 0 6 5 4 3 2 1 0 ntreal – p.22/41 /ν 1 | ) 1 − steps. j t ( N γ − x ) j with t ( 1 = ∆ γ − | | i Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo ) t j steps +1 X i t 0 ( N → γ x ∆ − ) i t steps for a SAW with ]= lim ( 2 γ , t | [0 N/ “length” be ﬁrst time after i . Then t length > t j t . Let -variation or fractal variation: 0 p

SAW vs SLE x> The endpoint of SAW with The point after SLE at a ﬁxed ∆ • • • Now compare the following Length is Let Stop when ntreal – p.23/41 2 1.5 midpoint of SAW endpoint of SAW SLE at fixed length Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo 1 Distance from origin 0.5

SAW vs SLE 0 1 0 1.8 1.6 1.4 1.2 0.8 0.6 0.4 0.2 ntreal – p.24/41 8 / 5 − ] 4 1 + 2 x [ 96 / 5 Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo + 1] 2 x [ 48 / . i 55 x )= x ≤ X ( P apply conformal map (random) of half plane to itself that

SAW endpoint and radial SLE? X=max of real part of points on walk. Do you get radial SLE (in the half plane) in the scaling limit? takes endpoint of SAW to Fixed length SAW gives a curve from boundary toIs interior. it related to radial SLE? Proposal: Test: Exact distribution is known: ntreal – p.25/41 2.5 2 SLE simulation Exact radial SLE SAW 200K steps 1.5 Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo X 1 SLE sim only has 5K samples, dx=0.01 0.5

Image of SAW is not radial SLE 0 1 0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 ntreal – p.26/41 , ): D → ′ D . ′ D Φ: (0)) ′ → D log(Φ . c D = on self-avoiding loops in the } ′ µ D Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo 6⊂ conformal map D, γ φ to loops inside ⊂ be simply connected. Let µ , γ D . ) φ γ ⊂ ( ′ D int . Then ∈ 0 ∈ 0 > :0 related by γ Let ′ (0) { ′ D ( Φ µ be simply connected, , µ , ′

4. Conformally invariant self-avoiding loops be the restriction of D D µ , D D µ is a measure on single loops, notmust ensembles be of an loops. inﬁnite measure. Then µ µ plane with following property Let Φ(0) = 0 Werner (building on work ofThere Lawler, is W an and essentially Lawler, unique Schramm, measure W. Proposition: Let ntreal – p.27/41 se loops. steps, shift so N 2 . ∞ and 0 Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo . . /a 0 1 − → and 0 . ) a Take SAW’s starting at origin with − z ( / is a loop passing through , lattice spacing ω

Bi-infinite SAW as self-avoiding loop )=1 →∞ z is a loop through ( φ N ) ω ( Let This walk φ Bi-infinite SAW: midpoint is at origin. Let But these loops go through two fixedHow points. is it related to Werner’s infinite measure? Probability measure on SAW gives probability measure on the ntreal – p.28/41 Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo

Bi-inﬁnite SAW as self-avoiding loop ntreal – p.29/41 lation, . 0 → area of interior, ǫ r dr )= ) γ ( w γ on shapes. ( 2 A d , A P ) } Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo ) π w γ dθ 2 ( − ) can be normalized. . γ γ ǫ int ( ( B iθ ∈ dP 0 re shapes = )= /ǫ, 1 γ ˆ γ (ˆ < dµ | γ With | restricted to µ ǫ< : , so γ { ∞

Werner’s measure on shapes = to deﬁne a probability measure < be the probability measure we get when ǫ ) µ ǫ P B B ( This gives a probability measure onLet shapes. Let µ Deﬁne two loops to bedilation equivalent and if rotation. they are related byCall a equivalence trans classes Easy propositon: Use ntreal – p.30/41 ? . P ) ′ D D 6⊂ ) ) w . } − , for ) ′ γ, w, θ γ ( γ ( D ( F (0)) contained in iθ ′ 6⊂ π int 0 dθ 2 ˆ γ ∈ Z D, re D, w log(Φ c w ⊂ ⊂ iff 2 Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo ) ) d ˆ γ γ , w )= (ˆ ) Z E γ − ( (ˆ ) int γ µ γ ( 1 ( ∈ int iθ A 0 ) ∈ a disc containing re γ ′ ( 1( :0 D , so r ) ˆ γ dP dr { w Z Z = − γ E ( )= )= iθ E ( re µ = γ, w, θ to be unit disc,

Bi-inﬁnite SAW as self-avoiding loop ( ˆ γ F D Take Test using the explicit formula Is the bi-inﬁnite SAW probability measure equal to Werner’s Recall where ntreal – p.31/41 1 0.9 0.8 0.7 Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo 0.6 0.5 0.4 0.3 0.2

Bi-inﬁnite SAW as self-avoiding loop 0.1 4 3 2 1 0 3.5 2.5 1.5 0.5 ntreal – p.32/41 1 0.9 0.8 0.7 0.6 Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo 0.5 0.4 0.3 0.2

Bi-infinite SAW as self-avoiding loop 0.1 1 1.04 1.02 0.98 0.96 ntreal – p.33/41 . µ om . . γ 0 } → /ǫ ǫ 1 which is invariant under r < dr . γ ) ) 2 γ γ ) w ( ( γ 2 ( d , λd d ) Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo π w can be normalized. dθ 2 )= − ) , ǫ < d ǫ,δ γ λγ γ ) ( ( ( B γ d d ( iθ . re dP γ < δd = | )= ˆ γ ) γ γ (ˆ ( can be any function of ) C dµ restricted to | With γ : ( µ d γ { , so = ∞ denote the center of mass of the interior of ǫ,δ be the diameter of < Confusion ) be the probability measure we get when B ) γ ) γ ( d ( ǫ,δ P d C B ( This gives another probability measure on shapes. Let More generally, µ Define There are other ways to get a probability measure on shapes fr Easy proposition: Let Let rotations and translations and ntreal – p.34/41 . P e? r dr r dr 2 ) ) w w γ γ 2 ( 2 ( d d d A Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo π π dθ 2 dθ 2 ? ) ) γ γ P ( ( 2 ) ) γ γ ( dP ( dP A d )= )= γ γ starting at origin, conditioned to end at origin and (ˆ (ˆ d P dµ , compare dµ ) w − γ ( iθ random walk re Bi-infinite SAW as self-avoiding loop = ˆ γ Which probability measure on shapes does bi-infiniteCan SAW you giv study this by simulations? Simulate (Brownian bridge). Take its outer boundary. This gives a probability measure on loops which is known to be Let Can you distinguish ntreal – p.35/41 Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo

’s ′ D

The ntreal – p.36/41 1.8 1 1.6 slit 1.4 disc at 0.15

formula) ) 1.2 γ ( crescent, center at 1.5 crescent, center at 2.0 crescent, center at 1.25 1 Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo /A 1 0.8 0.6 0.4 0.2

RW loop using area ( 0 1 1.1 0.9 1.05 0.95 Note vertical scale ntreal – p.37/41 1.8 1.6 slit

formula) 0.29555 2 1.4 disc at 0.15 ) γ ( 1.2 /d 1 crescent, center at 1.5 crescent, center at 2.0 crescent, center at 1.25 1 Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo 0.8 0.6 0.4 0.2

RW loop using diameter ( 0 0.3 0.32 0.31 0.29 0.28 0.27 ntreal – p.38/41 Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo ’s ′ D formula ) formula γ ( 2 ) using γ ( µ What about SAW? /Area /d 1 1 • • 200K steps in SAW Return to the bi-inﬁnite SAW. Look at the same ﬁve 77 CPU-days Look at ntreal – p.39/41 1.8 1 1.6 slit 1.4 disc at 0.15 formula) ) γ 1.2 ( crescent, center at 1.5 crescent, center at 2.0 crescent, center at 1.25 /A 1 Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo 1 0.8 0.6 0.4 0.2

SAW loop using area ( 0 1 1.1 0.9 1.05 0.95 ntreal – p.40/41 1.8 1.6 slit

formula) 0.2809 2 ) 1.4 disc at 0.15 γ ( /d 1.2 1 crescent, center at 1.5 crescent, center at 2.0 crescent, center at 1.25 1 Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo 0.8 0.6 0.4 0.2

SAW loop using diameter( 0 0.3 0.29 0.28 0.27 0.26 ntreal – p.41/41 int of adial . 3 / 8 . Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Mo i

Conclusions/Homework for the week Simulations of bond SAW agree with chordalSimulate SLE variable length SAW to check itSimulations agrees. of two mutually avoidingCardy-Gamsa SAW’s formulae. agree with Simulations indicate that fixed length SAW is not related to r Does SLE tell us anything about the distribution of the endpo What probability measure on shapesgive? does What the is bi-infinite the SAW RN derivativeCaution: with respect above is to hard P? to study by simulation. SLE by maping the endpoint to a fixed length SAW? • • • • • • •