Conformal invariance and covariance of the 2d self-avoiding walk

Transcription

1 Conformal invariance and covariance of the 2d self-avoiding walk Department of Mathematics, University of Arizona AMS Western Sectional Meeting, April 17, 2010 p.1/24

2 Outline Conformal invariance/covariance of random walk/brownian motion Definitions of self-avoiding walk (SAW) Conformal invariance/covariance of SAW Simulations of the SAW Lattice effects Conclusions, open questions Joint work with Greg Lawler, Ben Dyhr, Michael Gilbert, Shane Passon, Gabriel Moreno, Howard Cheng. AMS Western Sectional Meeting, April 17, 2010 p.2/24

3 Conformal invariance of Brownian motion What does this mean? Start a BM at the point, look at its exit point Probability it lies in red segment is conformally invariant. AMS Western Sectional Meeting, April 17, 2010 p.3/24

4 Conformal invariance of Brownian motion Assume boundary is smooth so that distribution of exit point has a density ρ(z) with respect to Lebesgue measure along the curve. This density is conformally covariant, not invariant. Let ρ D (z) be density for domain D. f conformal map on D. Then ρ D (z) = f (z) ρ f(d) (f(z)) If we fix a boundary point w and condition on the event that the walk exits at w, then this probability measure on curves is conformally invariant. AMS Western Sectional Meeting, April 17, 2010 p.4/24

5 Conformal invariance of Brownian motion Change of view. Fix a boundary point w and a boundary arc C. Start a BM at w, condition on the event that it exits D through C. Hitting distribution on C is conformally invariant. C w AMS Western Sectional Meeting, April 17, 2010 p.5/24

6 The SAW - grand canonical ensemble Let δ > 0 (the lattice spacing). We work on the lattice δz 2. Let D be a bounded domain and z, w D. Take all nearest neighbor walks in D from z to w which do not visit any site more than once. Weight a walk ω by β ω. The number of self-avoiding walks of length N in the full plane grows like β N c. Take β = β c. Normalize to get a probability measure. Try to let lattice spacing go to zero to get a probability measure on curves in D from z to w. If C is a boundary arc and w a boundary point we can also condsider SAW s starting at w and ending in C. Could also consider unbounded domain. AMS Western Sectional Meeting, April 17, 2010 p.6/24

7 The SAW - canonical ensemble Let D be an unbounded domain and z D. Think of the upper half plane with z = 0. Take all nearest neighbor walks in D which start at z, have N steps and do not visit any site more than once. Give them the uniform probability measure. Try to let N, then δ 0. If D is the uppper half plane, the N limit is known to exist. AMS Western Sectional Meeting, April 17, 2010 p.7/24

8 SAW - scaling limit AMS Western Sectional Meeting, April 17, 2010 p.8/24

9 Conformal invariance of SAW Physicists conjectured that SAW is conformally invariant. Lawler, Schramm and Werner gave a precise formulation: SAW between two fixed points is conformally invariant. They also conjectured that the scaling limit of the SAW is SLE 8/3. Monte Carlo simulations of the SAW support this conjecture. Past simulations have only been for canonical ensemble in the half plane or cut plane. It is not expected that hitting distribution is conformally invariant. The partition function point of view of SLE gives predictions for hitting distributions. AMS Western Sectional Meeting, April 17, 2010 p.9/24

10 SLE partition functions and hitting densities Partition Functions, Loop Measure, and Versions of SLE Lawler, Journal of Statistical Physics, 134, (2009) Return to normalization for grand canonical SAW. Call it N(D, z, w, δ) As δ 0, it is believed to go to zero as δ 2b. And lim δ 0 N(D, z, w, δ)δ 2b should exist. Call it C(D, z, w). This should be related to SLE partition functions which we denote as Z(D, z, w). SLE partition functions are conformally covariant: Z(D; z, w) = f (z) b f (w) b Z(f(D); f(z), f(w)) Is C(D, z, w) proportional to Z(D, z, w)? AMS Western Sectional Meeting, April 17, 2010 p.10/24

11 Conformal covariance of SAW C w 0 1 Conjecture: ρ D,C (z) = c f (z) 5/8 ρ f(d),f(c) (f(z)) ρ H,(1, ) (x) = c x 5/4 AMS Western Sectional Meeting, April 17, 2010 p.11/24

12 Monte Carlo tests Easy to simulate SAW in half plane from 0 to. Hard to simulate SAW between two finite points. AMS Western Sectional Meeting, April 17, 2010 p.12/24

13 Conditioning: SAW in half plane SAW in strip How is half plane SAW related to SAW in strip? Consider ordinary random walk conditioned to stay in upper half plane. Stop it when it hits line of height y. This is same as RW in the strip. Do the same thing for SAW in H and you don t get SAW in strip. Theorem Fix a positive integer y. Condition on the event that SAW intersects the line y 1/2 once. Only consider the walk up to height y Then it has the distribution of the SAW in a strip Conjecture: ρ(x) = c [ cosh ( )] 5/4 πx 2y AMS Western Sectional Meeting, April 17, 2010 p.13/24

14 Test of density conjecture for the strip SLE SAW Cumulative distribution x AMS Western Sectional Meeting, April 17, 2010 p.14/24

15 More general bridges or cut curves Bridges: intersect horizontal line once Generalize: horizontal line another curve, e.g., semicircles in H Look for SAW s that intersect the curve once SLE partition functions predict ρ(θ) = [sin(θ)] 5/8 [sin(θ)] 5/8. AMS Western Sectional Meeting, April 17, 2010 p.15/24

16 Lattice effects SAW in full plane Condition to hit a circle exactly once. AMS Western Sectional Meeting, April 17, 2010 p.16/24

17 Lattice effects R=0.1 R=0.2 R=0.3 R= Deviation from uniform cdf Polar angle AMS Western Sectional Meeting, April 17, 2010 p.17/24

18 Lattice effects Square Triangular Hexagonal Deviation from uniform cdf Polar angle AMS Western Sectional Meeting, April 17, 2010 p.18/24

19 Lattice effects Conjecture: There is function l(θ) (depends on the type of lattice) such that ρ D (z) = c f (z) 5/8 ρ f(d) (f(z)) l(θ(z)) where θ(z) is the angle of the tangent to the boundary at z. Note: l(θ) only depends on the angle, not on the domain. It is a local effect. C w AMS Western Sectional Meeting, April 17, 2010 p.19/24

20 Lattice effects Conjecture: Lattice effect is a factor l(θ) where θ is the angle between the boundary and the lattice directions. AMS Western Sectional Meeting, April 17, 2010 p.20/24

21 Lattice effect function Lattice effect function l(theta) - square lattice theta AMS Western Sectional Meeting, April 17, 2010 p.21/24

22 Full plane hitting circle Full plane SAW hitting circle: difference of simulation and conjecture (100K steps, 31M samples) Without l(theta) correction 0 With l(theta) correction theta AMS Western Sectional Meeting, April 17, 2010 p.22/24

23 Half plane hitting semi-circle Half plane hitting semicircle: difference of simulation and conjecture (100K steps, 73M samples) Without l(theta) correction 0 With l(theta) correction theta AMS Western Sectional Meeting, April 17, 2010 p.23/24

24 Conclusions, questions You can get SAW in a strip from SAW in H by conditioning SLE partition function prediction for hitting density for strip agree Simluations of SAW in a strip agree with SLE SLE partition function prediction for hitting densities for other geometries have lattice effects Lattice effect can be computed a priori - local effect There are no rigorous results on the scaling limit of SAW AMS Western Sectional Meeting, April 17, 2010 p.24/24