monte carlo studies of self-avoiding walks and loopstgk/talks/montreal.pdfoutline • 0. review: def...

Monte Carlo studies of Self-Avoiding Walksand 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

PrefaceThe organizers wrote

“We would like to encourage you to speak on ongoing or recentdevelopments rather than to deliver a lecture that you have alreadygiven on several occasions.”

I have not given this talk before.





I will not give this talk again.





I will not give this talk again.

I will mainly talk about ongoing work with an emphasis on things I donot understand.


Outline• 0. Review: Def of SAW, conjectured relation to SLE8/3• 1. Bond avoiding random walk• 2. Two saw’s - comparison with Cardy/Gamsa formula• 3. Distribution of points on SAW and SLE

interior point vs. endpointSAW vs. SLE

• 4. Bi-infinite SAW as a self-avoiding loopcomparison with Werner’s measure on self-avoiding loops


Definition of SAWTake all N step, nearest neighbor walks in the upper half plane,starting at the origin which do not visit any site more than once.

Give them the uniform probability measure.

Let N → ∞. Then let lattice spacing go to zero.


Relation to SLE8/3Do this in the upper half plane

Conjecture (LSW) : the scaling limit of the SAW is chordal SLE8/3

Simulations of SAW support the conjecture

SAW in other geometries typically requires a variable number of stepsN with a weight βN .

Note: All simulations supporting the conjecture have been for SAW’swith a fixed number of steps.


1. Bond-avoiding walk

Take all nearest neighbor walks of length N that self-avoid in the sensethat they do not traverse the same bond more than once.

Large loops are allowed, but entropically suppressed.

Everyone expects this model to have same scaling limit as usual SAW.

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1700 1720 1740 1760 1780 1800Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.6/41

Bond and site avoiding SAW’sPicture of two bond avoiding walks and two site avoiding walks

"1""2""3""4"


Bond and site avoiding SAW’sBlow up of previous picture showing small loops in bond SAW

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Answer: 1=bond, 2=bond, 3=site, 4=siteTom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.8/41

Probability of passing rightSchramm gave an explicit formula for the probability the SLE curvepasses to the right of a fixed point. Difference is about 0.1%.

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prob

abili

ty o

f pas

sing

rig

ht

theta/pi

bond SAW simulationSchramm’s formula


RV testsLawler, Schramm and Werner gave an explicit formula for theprobability of certain events for SLE8/3.

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Y

X


Test usingX RV

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CD

F’s

Distance to 1

bond SAW simulationExact LSW result


Test usingX RV

-0.025

-0.02

-0.015

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-0.005

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0.005

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diffe

renc

e in

CD

F’s

Distance to 1


Test usingY RV

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CD

F’s

min height along vertical line

bond SAW simulationLSW exact result


Test usingY RV

-0.01

-0.005

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diffe

renc

e of

CD

F’s

min height along vertical line


2. Two SAW’sConsider two SAW’s in the upper half plane both starting at the originwith the condition that they do not intersect each other.

Uniform probability measure


Two SLE’sCardy and Gamsa considered two SLE curves starting at the origin.

Used boundary CFT to derive the probabilities that a given point is leftof both curves, in the middle of the curves or to the right of bothcurves. For κ = 8/3 their formulae are

Let t = cot(θ).

Pleft =−2t(13 + 15t2) + (3π − 6 arctan(t))(1 + 6t2 + 5t4)

30π(1 + t2)2

Pmiddle =4

5(1 + t2)

Pright =2t(13 + 15t2) + (3π + 6 arctan(t))(1 + 6t2 + 5t4)

30π(1 + t2)2


SAW simulation vs. CFT formula

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0 0.2 0.4 0.6 0.8 1theta/pi

SAW’s

exact


Differences - SAW’s vs exact

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right

middleleft


3. Distribution of points on SAW and SLESAW with N steps is not the same as the first N steps of a SAW with2N steps.

SLE on time interval [0, 1] is the same as SLE on [0, 2] restricted to[0, 1].

"100K steps""200K steps"


SAW vs SLEWe consider the distribution of the random variable which is thedistance from the origin to various points on the random curves.

We compare the following• The endpoint of SAW with N steps• The point after N/2 steps for a SAW with N steps.

• SLE at time (half-plane capacity) 1.

All RV’s are normalized to have mean 1.


SAW vs SLE

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0 0.5 1 1.5 2Distance from origin

midpoint of SAWendpoint of SAW

SLE at fixed capacity


SAW vs SLENow compare the following

• The endpoint of SAW with N steps• The point after N/2 steps for a SAW with N steps.

• SLE at a fixed “length”

Length is p-variation or fractal variation:

Let ∆x > 0. Let ti be first time after ti−1 with

|γ(ti) − γ(ti+1)| = ∆x

Stop when tj > t. Then

length[0, t] = lim∆x→0

∑j

|γ(tj) − γ(tj−1)|1/ν


SAW vs SLE

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midpoint of SAWendpoint of SAW

SLE at fixed length


SAW endpoint and radial SLE?Fixed length SAW gives a curve from boundary to interior.

Is it related to radial SLE?

Proposal: apply conformal map (random) of half plane to itself thattakes endpoint of SAW to i.

Do you get radial SLE (in the half plane) in the scaling limit?

Test: X=max of real part of points on walk.

Exact distribution is known:

P (X ≤ x) = x55/48 [x2 + 1]5/96 [x2 +1

4]−5/8


Image of SAW is not radial SLE

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X

SLE sim only has 5K samples, dx=0.01

Exact radial SLESLE simulation

SAW 200K steps


4. Conformally invariant self-avoiding loops

Werner (building on work of Lawler, W and Lawler, Schramm, W.):There is an essentially unique measure µ on self-avoiding loops in theplane with following property

Let D,D′ be simply connected, φ conformal map D → D′.

Let µD be the restriction of µ to loops inside D.

Then µD, µD′ related by φ.

µ is a measure on single loops, not ensembles of loops.

µ must be an infinite measure.

Proposition: Let 0 ∈ D′ ⊂ D be simply connected. Let Φ : D′ → D,Φ(0) = 0, Φ′(0) > 0. Then

µ({γ : 0 ∈ int(γ), γ ⊂ D, γ 6⊂ D′} = c log(Φ′(0))


Bi-infinite SAW as self-avoiding loop

Bi-infinite SAW: Take SAW’s starting at origin with 2N steps, shift somidpoint is at origin.

Let N → ∞, lattice spacing → 0.

This walk ω is a loop passing through 0 and ∞.

Let φ(z) = 1/(z − a).

φ(ω) is a loop through 0 and −1/a.

Probability measure on SAW gives probability measure on these loops.

But these loops go through two fixed points.

How is it related to Werner’s infinite measure?


Werner’s measure on shapes

Define two loops to be equivalent if they are related by a translation,dilation and rotation.

Call equivalence classes shapes.

Use µ to define a probability measure P on shapes.

Let Bǫ = {γ : ǫ < |γ| < 1/ǫ, 0 ∈ int(γ)}

µ(Bǫ) < ∞, so µ restricted to Bǫ can be normalized.

This gives a probability measure on shapes.

Let P be the probability measure we get when ǫ → 0.

Easy propositon: With γ̂ = reiθ(γ − w), A(γ) = area of interior,

dµ(γ̂) = dP (γ)dθ

2π

d2w

A(γ)

dr

r


Bi-infinite SAW as self-avoiding loopIs the bi-infinite SAW probability measure equal to Werner’s P?

Test using the explicit formula µ(E) = c log(Φ′(0)), for

E = {γ̂ : 0 ∈ int(γ̂), γ̂ ⊂ D, γ̂ 6⊂ D′}

Recall γ̂ = reiθ(γ − w), so 0 ∈ int(γ̂) iff w ∈ int(γ).

µ(E) =

∫dP (γ)

1

A(γ)

∫d2w

∫dθ

2πF (γ, w, θ)

where

F (γ, w, θ) =

∫dr

r1(reiθ(γ − w) ⊂ D, reiθ(γ − w) 6⊂ D′)

Take D to be unit disc, D′ a disc containing 0 contained in D.



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Confusion

There are other ways to get a probability measure on shapes from µ.

Let C(γ) denote the center of mass of the interior of γ.

Let d(γ) be the diameter of γ.

Define Bǫ,δ = {γ : |C(γ)| < δd(γ), ǫ < d(γ) < 1/ǫ}

µ(Bǫ,δ) < ∞, so µ restricted to Bǫ,δ can be normalized.

This gives another probability measure on shapes.

Let P d be the probability measure we get when ǫ → 0.

More generally, d(γ) can be any function of γ which is invariant underrotations and translations and d(λγ) = λd(γ).Easy proposition: With γ̂ = reiθ(γ − w),

dµ(γ̂) = dP d(γ)dθ

2π

d2w

d(γ)2dr

r


Bi-infinite SAW as self-avoiding loopWhich probability measure on shapes does bi-infinite SAW give?

Can you study this by simulations?

Simulate random walk starting at origin, conditioned to end at origin(Brownian bridge). Take its outer boundary.

This gives a probability measure on loops which is known to be P .

Can you distinguish P and A(γ)d(γ)2 P ?

Let γ̂ = reiθ(γ − w), compare

dµ(γ̂) = dP (γ)dθ

2π

d2w

A(γ)

dr

r

dµd(γ̂) = dP (γ)dθ

2π

d2w

d(γ)2dr

r


TheD′’s


RW loop using area (1/A(γ) formula)Note vertical scale

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crescent, center at 1.25crescent, center at 1.5crescent, center at 2.0

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RW loop using diameter (1/d(γ)2 formula)

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What about SAW?

Return to the bi-infinite SAW.

Look at the same five D′’s

Look at µ using• 1/Area(γ) formula

• 1/d(γ)2 formula

200K steps in SAW

77 CPU-days


SAW loop using area (1/A(γ) formula)

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SAW loop using diameter(1/d(γ)2 formula)

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Conclusions/Homework for the week

• Simulations of bond SAW agree with chordal SLE8/3.

• Simulate variable length SAW to check it agrees.• Simulations of two mutually avoiding SAW’s agree with

Cardy-Gamsa formulae.• Simulations indicate that fixed length SAW is not related to radial

SLE by maping the endpoint to i.• Does SLE tell us anything about the distribution of the endpoint of

a fixed length SAW?• What probability measure on shapes does the bi-infinite SAW

give? What is the RN derivative with respect to P?• Caution: above is hard to study by simulation.


PrefaceOutlineDefinition of SAWRelation to SLE$_{8/3}$ed 1. Bond-avoiding walkBond and site avoiding SAW'sBond and site avoiding SAW'sProbability of passing rightRV testsTest using $X$ RVTest using $X$ RVTest using $Y$ RVTest using $Y$ RVed 2. Two SAW'sTwo SLE'sSAW simulation vs. CFT formulaDifferences - SAW's vs exacted 3. Distribution of points on SAW and SLESAW vs SLE SAW vs SLE SAW vs SLE SAW vs SLE SAW endpoint and radial SLE?Image of SAW is not radial SLEed 4. Conformally invariant self-avoiding loopsBi-infinite SAW as self-avoiding loopBi-infinite SAW as self-avoiding loopWerner's measure on shapesBi-infinite SAW as self-avoiding loopBi-infinite SAW as self-avoiding loopBi-infinite SAW as self-avoiding loopConfusionBi-infinite SAW as self-avoiding loopThe $D^prime $'sRW loop using area ($1/A(gamma )$formula)RW loop using diameter ($1/d(gamma )^2$formula)What about SAW?SAW loop using area ($1/A(gamma )$formula)SAW loop using diameter($1/d(gamma )^2$formula)Conclusions/Homework for the week

monte carlo studies of self-avoiding walks and loopstgk/talks/montreal.pdfoutline • 0. review: def...

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