The Length of an SLE - Monte Carlo Studies

Transcription

1 The Length of an SLE - Monte Carlo Studies Department of Mathematics, University of Arizona Supported by NSF grant DMS tgk Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.1/29

2 The Self-Avoiding Walk 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. Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.2/29

3 The Self-Avoiding Walk 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. Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.2/29

4 Natural parameterization of SAW SAW has a natural parameterization. Let W(n) be infinite SAW on unit lattice. Define ω(t) = lim n n ν W(nt), E ω(t) 2 = c t 2ν Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.3/29

5 Natural parameterization of SAW SAW has a natural parameterization. Let W(n) be infinite SAW on unit lattice. Define ω(t) = lim n n ν W(nt), SAW: 100,000 steps, 40 segments of 2,500 steps E ω(t) 2 = c t 2ν Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.3/29

6 Capacity parameterization of SLE SLE is usually parameterized by half-plane capacity. If we divide it into segments of equal change in capacity, we get Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.4/29

7 Capacity parameterization of SLE SLE is usually parameterized by half-plane capacity. If we divide it into segments of equal change in capacity, we get Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.4/29

8 Fractal variation parameterization of SLE If we take the same SLE curves from previous slide and parameterize them by p-variation and divide it into segments of equal variation, we get Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.5/29

9 Fractal variation parameterization of SLE If we take the same SLE curves from previous slide and parameterize them by p-variation and divide it into segments of equal variation, we get Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.5/29

10 Why you want to reparameterize Consider the SAW. It is believed to be SLE(8/3). What does this mean? If we consider the SAW scaling limit curves and SLE curves modulo parameterization, then they have the same distribution. If we use the natural parameterization for SAW and the capacity parameterization for SLE, the parameterized curves do not have the same distribution. If you use capacity to parameterize the SAW, then the SAW curves and the SLE curves should have the same distribution as parameterized curves. Our goal is to reparameterize SLE so it agrees with SAW with its natural parameterization. Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.6/29

11 How to reparameterize SLE Use pth variation with p = 1/ν: Let 0 = t n 0 < t n 1 < t n 2 < t n k n = t be sequence of partitions of [0, t]. fvar(γ[0, t]) = lim γ(t n n j ) γ(t n j 1) 1/ν j With ν = 1/2 this is the quadratic variation. For Brownian motion it is non-random and proportional to t. Problem: definition of pth variation depends on parameterization If you compute pth variation for SAW using capacity parameterization and the natural parameterization, you do not get the same result. Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.7/29

12 Better definition of fractal variation Let x > 0. Let t i be first time after t i 1 with γ(t i ) γ(t i+1 ) = x Then γ(t j ) γ(t j 1 ) 1/ν = n x 1/ν j where n depends on the curve. Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.8/29

13 Better definition of fractal variation Let x > 0. Let t i be first time after t i 1 with γ(t i ) γ(t i+1 ) = x Then γ(t j ) γ(t j 1 ) 1/ν = n x 1/ν j where n depends on the curve. Conjecture: For the scaling limit of a discrete model (LERW, SAW, Ising, percolation), the fractal variation exists and fvar(ω[0, t]) = ct Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.8/29

14 Better definition of fractal variation Let x > 0. Let t i be first time after t i 1 with γ(t i ) γ(t i+1 ) = x Then γ(t j ) γ(t j 1 ) 1/ν = n x 1/ν j where n depends on the curve. Conjecture: For the scaling limit of a discrete model (LERW, SAW, Ising, percolation), the fractal variation exists and fvar(ω[0, t]) = ct Key point: The 1/ν variation of ω[0, t] is not random. (LLN) Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.8/29

15 LERW Fractal Variation Compute fractal variation of LERW s of fixed length dt=0.01 dt=0.005 dt=0.002 dt=0.001 LERW fractal variation, 100K steps Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.9/29

16 LERW Fractal Variation - zoomed Compute fractal variation of LERW s of fixed length dt=0.01 dt=0.005 dt=0.002 dt=0.001 LERW fractal variation, 100K steps Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.10/29

17 LERW Fractal Variation If dt is too small dt=0.01 dt=0.005 dt=0.002 dt=0.001 dt= dt= dt= LERW fractal variation, 100K steps Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.11/29

18 SAW Fractal Variation Self-avoiding walks with 1,000,000 steps SAW, fractal variation, 1000K steps dt=0.01 dt=0.005 dt=0.002 dt=0.001 dt= dt= Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.12/29

19 Ising Fractal Variation Interface of Ising model in 1200 by 400 box with Dobrushin boundary conditions Ising interface, fractal variation, 1200 x 400 dt=0.01 dt=0.005 dt=0.002 dt= Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.13/29

20 Percolation Fractal Variation Percolation exploration process in half plane, 4,000,000 steps Percolation, fractal variation, 4000K steps dt=0.01 dt=0.005 dt=0.002 dt=0.001 dt= Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.14/29

21 SLE vs Discrete Models as Parameterized Curves The theorems/conjectures of the form LERW=SLE 2, SAW=SLE 8/3, Ising=SLE 3, percolation=sle 6 are typically statements about paths modulo parametrization or statements where the discrete model is parameterized by capacity. These correspondences should hold if we use the natural parameterization for the discrete model and the fractal variation to parameterize the SLE. To test this we must simulate SLE as well as the discrete models. Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.15/29

22 How to simulate SLE Discretize time, but not space: 0 < t 1 < t 2 < < t n = t. Sample BM at these times. Approximate the BM path by a function which agrees at the t i and in between is interpolated in such a way that the Loewner equation has an explicit solution, e.g., c t Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.16/29

23 SLE Simulation subtleties f t = gt 1 is now given by a composition of n explicit conformal maps. Points on SLE are given by z k = f k f 2 f 1 (0) SLE(6) with uniform time discretization - 20,000 steps Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.17/29

24 SLE Simulation subtleties Better to use a non-uniform time discretization, depending on the BM sample. When step of SLE is too large, sample the BM at a shorter time scale for that time interval using Brownian bridge. SLE(6) with adaptive time discretization - 20,000 steps Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.18/29

25 SLE Simulation subtleties Points on SLE are given by z k = f k f 2 f 1 (0) Time needed to compute n points on SLE is O(n 2 ). There is a trick to make it O(n p ) with p 1.4. z k = f k f lb+1 (f lb f lb+2 f lb+1 ) (f 2b f b+2 f b+1 ) (f b f 2 f 1 )(0) Approximate f i by Laurent series about. Then Laurent series of block function can be computed. Use it when you can. Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.19/29

26 SLE vs Discrete For LERW, SAW, percolation in upper half plane, we fix a semicircle of some radius. Compute fractal variation of curve until it hits semicircle. Then look at point that is midway between this point and origin. Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.20/29

27 LERW vs SLE(2) in half plane: X Distribution of x-coordinate of midpoint. 1 LERW-chordal half plane SLE(2) SLE(2) using capacity Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.21/29

28 LERW vs SLE(2) in half plane: Y Distribution of y-coordinate of midpoint. 1 LERW - chordal half plane SLE(2) SLE(2) using capacity Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.22/29

29 SAW vs SLE(8/3) in half plane: X Distribution of x-coordinate of midpoint. 1 SAW half plane SLE(8/3) Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.23/29

30 SAW vs SLE(8/3) in half plane: Y Distribution of x-coordinate of midpoint. 1 SAW SLE(8/3) Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.24/29

31 Ising vs SLE(3) in half plane: X Distribution of x-coordinate of midpoint Ising SLE(3) Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.25/29

32 Ising vs SLE(3) in half plane: Y Distribution of y-coordinate of midpoint. 1 Ising SLE(3) Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.26/29

33 Percolation vs SLE(6) in half plane: X Distribution of x-coordinate of midpoint. 1 "perc_0.50_x" "sle_6._0.01_0.50_x" Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.27/29

34 Percolation vs SLE(6) in half plane: Y Distribution of y-coordinate of midpoint. 1 "perc_0.50_y" "sle_6._0.01_0.50_y" Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.28/29

35 Conclusions/conjectures/homework For discrete models the fractal variation exists and is proportional to the natural parameterization. Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.29/29

36 Conclusions/conjectures/homework For discrete models the fractal variation exists and is proportional to the natural parameterization. Monte Carlo simulations of LERW, SAW, Ising spin interface and percolation exploration support this. Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.29/29

37 Conclusions/conjectures/homework For discrete models the fractal variation exists and is proportional to the natural parameterization. Monte Carlo simulations of LERW, SAW, Ising spin interface and percolation exploration support this. For SLE the fractal variation exists. Can you prove this? Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.29/29

38 Conclusions/conjectures/homework For discrete models the fractal variation exists and is proportional to the natural parameterization. Monte Carlo simulations of LERW, SAW, Ising spin interface and percolation exploration support this. For SLE the fractal variation exists. Can you prove this? If we use fractal variation to parameterize SLE, then it agrees as a parameterized curve with the discrete model using its natural time as the parameterization. Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.29/29