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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/e tgk Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.1/41 Preface The organizers wrote “We would like to encourage you to speak on ongoing or recent developments rather than to deliver a lecture that you have already given on several occasions.” I have not given this talk before. Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.2/41 Preface The organizers wrote “We would like to encourage you to speak on ongoing or recent developments rather than to deliver a lecture that you have already given on several occasions.” I have not given this talk before. I will not give this talk again. Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.2/41 Preface The organizers wrote “We would like to encourage you to speak on ongoing or recent developments rather than to deliver a lecture that you have already given on several occasions.” I have not given this talk before. I will not give this talk again. I will mainly talk about ongoing work with an emphasis on things I do not understand. Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.2/41 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. endpoint SAW vs. SLE • 4. Bi-inﬁnite SAW as a self-avoiding loop comparison with Werner’s measure on self-avoiding loops Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.3/41 Deﬁnition of SAW Take 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. Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.4/41 Relation to SLE8/3 Do 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 steps N with a weight β N . Note: All simulations supporting the conjecture have been for SAW’s with a ﬁxed number of steps. Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.5/41 1. Bond-avoiding walk Take all nearest neighbor walks of length N that self-avoid in the sense that 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. 1220 1200 1180 1160 1140 1120 1700 1720 1740 1760 1780 1800 Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.6/41 Bond and site avoiding SAW’s Picture of two bond avoiding walks and two site avoiding walks "1" "2" "3" "4" Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.7/41 Bond and site avoiding SAW’s Blow up of previous picture showing small loops in bond SAW 5640 5620 5600 5580 5560 5540 5520 5500 5480 5460 650 700 750 800 850 900 Answer: 1=bond, 2=bond, 3=site, 4=site Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.8/41 Probability of passing right Schramm gave an explicit formula for the probability the SLE curve passes to the right of a ﬁxed point. Difference is about 0.1%. 1 0.8 probability of passing right 0.6 0.4 0.2 bond SAW simulation Schramm’s formula 0 0 0.2 0.4 0.6 0.8 1 theta/pi Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.9/41 RV tests Lawler, Schramm and Werner gave an explicit formula for the probability of certain events for SLE8/3 . Y X 0 Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.10/41 Test using X RV 1 bond SAW simulation Exact LSW result 0.8 0.6 CDF’s 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Distance to 1 Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.11/41 Test using X RV 0.005 0 difference in CDF’s -0.005 -0.01 -0.015 -0.02 -0.025 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Distance to 1 Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.12/41 Test using Y RV 1 bond SAW simulation 0.9 LSW exact result 0.8 0.7 0.6 CDF’s 0.5 0.4 0.3 0.2 0.1 0 0 2 4 6 8 10 12 14 16 18 20 min height along vertical line Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.13/41 Test using Y RV 0.01 0.005 difference of CDF’s 0 -0.005 -0.01 0 2 4 6 8 10 12 14 16 18 20 min height along vertical line Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.14/41 2. Two SAW’s Consider two SAW’s in the upper half plane both starting at the origin with the condition that they do not intersect each other. Uniform probability measure Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.15/41 Two SLE’s Cardy and Gamsa considered two SLE curves starting at the origin. Used boundary CFT to derive the probabilities that a given point is left of both curves, in the middle of the curves or to the right of both curves. For κ = 8/3 their formulae are Let t = cot(θ). −2t(13 + 15t2 ) + (3π − 6 arctan(t))(1 + 6t2 + 5t4 ) Plef t = 30π(1 + t2 )2 4 Pmiddle = 5(1 + t2 ) 2t(13 + 15t2 ) + (3π + 6 arctan(t))(1 + 6t2 + 5t4 ) Pright = 30π(1 + t2 )2 Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.16/41 SAW simulation vs. CFT formula 1 SAW’s exact 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 theta/pi Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.17/41 Differences - SAW’s vs exact 0.004 middle 0.003 left right 0.002 0.001 0 -0.001 -0.002 -0.003 -0.004 0 0.2 0.4 0.6 0.8 1 theta/pi Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.18/41 3. Distribution of points on SAW and SLE SAW with N steps is not the same as the ﬁrst N steps of a SAW with 2N steps. SLE on time interval [0, 1] is the same as SLE on [0, 2] restricted to [0, 1]. "100K steps" "200K steps" Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.19/41 SAW vs SLE We consider the distribution of the random variable which is the distance 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. Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.20/41 SAW vs SLE 6 midpoint of SAW endpoint of SAW 5 SLE at fixed capacity 4 3 2 1 0 0 0.5 1 1.5 2 Distance from origin Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.21/41 SAW vs SLE Now 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 |γ(tj ) − γ(tj−1 )|1/ν ∆x→0 j Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.22/41 SAW vs SLE 1.8 midpoint of SAW 1.6 endpoint of SAW SLE at fixed length 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 Distance from origin Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.23/41 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 that takes 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: 1 P (X ≤ x) = x55/48 [x2 + 1]5/96 [x2 + ]−5/8 4 Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.24/41 Image of SAW is not radial SLE SLE sim only has 5K samples, dx=0.01 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 Exact radial SLE SLE simulation SAW 200K steps 0 0 0.5 1 1.5 2 2.5 X Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.25/41 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 the plane 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 ⊂ D′ } = c log(Φ′ (0)) Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.26/41 Bi-inﬁnite SAW as self-avoiding loop Bi-inﬁnite SAW: Take SAW’s starting at origin with 2N steps, shift so midpoint 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? Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.27/41 Bi-inﬁnite SAW as self-avoiding loop Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.28/41 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 γ = reiθ (γ − w), A(γ) = area of interior, ˆ dθ d2 w dr γ dµ(ˆ ) = dP (γ) 2π A(γ) r Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.29/41 Bi-inﬁnite SAW as self-avoiding loop Is the bi-inﬁnite SAW probability measure equal to Werner’s P ? Test using the explicit formula µ(E) = c log(Φ′ (0)), for E = {ˆ : 0 ∈ int(ˆ ), γ ⊂ D, γ ⊂ D′ } γ γ ˆ ˆ Recall γ = reiθ (γ − w), so 0 ∈ int(ˆ ) iff w ∈ int(γ). ˆ γ 1 dθ µ(E) = dP (γ) d2 w F (γ, w, θ) A(γ) 2π where dr F (γ, w, θ) = 1(reiθ (γ − w) ⊂ D, reiθ (γ − w) ⊂ D′ ) r Take D to be unit disc, D′ a disc containing 0 contained in D. Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.30/41 Bi-inﬁnite SAW as self-avoiding loop 4 3.5 3 2.5 2 1.5 1 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.31/41 Bi-inﬁnite SAW as self-avoiding loop 1.04 1.02 1 0.98 0.96 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.32/41 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 under rotations and translations and d(λγ) = λd(γ). Easy proposition: With γ = reiθ (γ − w), ˆ d dθ d2 w dr γ dµ(ˆ ) = dP (γ) 2π d(γ)2 r Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.33/41 Bi-inﬁnite SAW as self-avoiding loop Which 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 . A(γ) Can you distinguish P and d(γ)2 P ? Let γ = reiθ (γ − w), compare ˆ dθ d2 w dr γ dµ(ˆ ) = dP (γ) 2π A(γ) r dθ d2 w dr dµd (ˆ ) = dP (γ) γ 2π d(γ)2 r Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.34/41 The D′ ’s Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.35/41 RW loop using area (1/A(γ) formula) Note vertical scale 1.1 disc at 0.15 slit crescent, center at 1.25 crescent, center at 1.5 crescent, center at 2.0 1.05 1 1 0.95 0.9 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.36/41 RW loop using diameter (1/d(γ)2 formula) disc at 0.15 0.32 slit crescent, center at 1.25 crescent, center at 1.5 crescent, center at 2.0 0.31 0.29555 0.3 0.29 0.28 0.27 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.37/41 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 Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.38/41 SAW loop using area (1/A(γ) formula) 1.1 disc at 0.15 slit crescent, center at 1.25 crescent, center at 1.5 crescent, center at 2.0 1.05 1 1 0.95 0.9 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.39/41 SAW loop using diameter(1/d(γ)2 formula) disc at 0.15 slit crescent, center at 1.25 0.3 crescent, center at 1.5 crescent, center at 2.0 0.2809 0.29 0.28 0.27 0.26 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.40/41 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. Tom Kennedy MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.41/41