Testing for SLE using the driving process

Transcription

1 Testing for SLE using the driving process Department of Mathematics, University of Arizona Supported by NSF grant DMS e tgk Testing for SLE, 13rd Itzykson Conference PUZZLES OF GROWTH, June 9-11, Saclay, France p.1/36

2 Outline Computing the Loewner driving process of a random curve - basics Computing the Loewner driving process of a random curve - faster Testing for SLE - stretched LERW, SAW, percolation Gradient-gradient percolation Conclusions, future work C++ code implementing faster algorithm is available at e tgk Go to the ENRAGE summer school talk at the bottom Testing for SLE, 13rd Itzykson Conference PUZZLES OF GROWTH, June 9-11, Saclay, France p.2/36

3 Introduction Let γ(t) be simple (non-intersecting) curve in upper half plane H starting at 0. g t (z) is conformal map from H \ γ[0, t] onto H. Loewner equation: g t (z) = z + 2t z + O( 1 z 2) g t (z) t = 2 g t (z) U t U t is real valued, called driving function Random γ stochastic driving process U t. SLE is when U t = κb t, B t standard Brownian motion. Testing for SLE, 13rd Itzykson Conference PUZZLES OF GROWTH, June 9-11, Saclay, France p.3/36

4 Motivation Given a random curve in H, starting at 0, is it SLE? Test if it is SLE by testing if U t is a Brownian motion? Domain walls in spin glass ground states - Amoruso, Hartman, Hastings, and Moore and - Bernard, Le Doussal, and Middleton Certain isolines in two-dimensional turbulence - Bernard, Boffetta, Celani and Falkovich Nodal lines in quantum chaotic systems - Bogomolny, Dubertrand, Schmit Look at models which we know are not SLE and those which are How good a test is this? What exactly should you test? Testing for SLE, 13rd Itzykson Conference PUZZLES OF GROWTH, June 9-11, Saclay, France p.4/36

5 Computing U t We find it more convenient to work with the conformal map h t (z) = g t (z) U t h t : H \ γ[0, t] H, sends tip γ(t) to the origin. The value of the driving function at t is minus the constant term in the Laurent expansion of h t about : h t (z) = z U t + 2t z + O( 1 z 2), Testing for SLE, 13rd Itzykson Conference PUZZLES OF GROWTH, June 9-11, Saclay, France p.5/36

6 Sequence of compositions Let γ be a curve in H starting at 0. Let z 0, z 1,,z n be points along the curve with z 0 = 0. In many applications these are lattice sites. Let 2t k be capacity of curve up to z k. h tk can be written as a sequence of compostions: h tk = h k h k 1 h 2 h 1 h tk sends z k to 0, z k+1 to a point w k+1 near the origin. Let γ k+1 be a short simple curve that ends at w k+1, e.g., tilted slit or vertical slit. Testing for SLE, 13rd Itzykson Conference PUZZLES OF GROWTH, June 9-11, Saclay, France p.6/36

7 Sequence of compositions - cont. Choose γ k+1 so that the conformal map that unzips it is explicitly know. Let h k+1 be this conformal map. Tilted slits: angle, length Vertical slits: length, displacement from origin Circular arcs: angle, radius h k+1 has two real degrees of freedom, determined by w k+1. Testing for SLE, 13rd Itzykson Conference PUZZLES OF GROWTH, June 9-11, Saclay, France p.7/36

8 Finding the driving function Let 2 t i be capacity of h i Let x i be final value of the driving function for h i. So h i (z) = z x i + 2 t i z + O( 1 z 2 ) Then h k h k 1 h 1 (z) = z U tk + 2t k z + O( 1 z 2) where t k = k t i, U tk = i=1 k x i i=1 Driving function of the curve is sum of driving functions of the elementary conformal maps h i. Testing for SLE, 13rd Itzykson Conference PUZZLES OF GROWTH, June 9-11, Saclay, France p.8/36

9 A faster zipper Single w k+1 takes O(k) time. All the w k+1 takes O(N 2 ). w k+1 = h k h k 1 h 1 (z k+1 ) Group functions we are composing into blocks. H j = h jb h jb 1 h (j 1)b+2 h (j 1)b+1 With k = mb + r with 0 r < b, w k+1 = h mb+r h mb+r 1 h mb+1 H m H m 1 H 1 (z k+1 ) Approximate composition within a block using a series. Compute this approximation to H j just once. Don t alway use the approximation to H j. Testing for SLE, 13rd Itzykson Conference PUZZLES OF GROWTH, June 9-11, Saclay, France p.9/36

10 How fast is it? Unzip a SAW. Plot time (in seconds) vs. N, number of step. Optimize block size b Lines have slope 2 and e without approximation with approximation Time (secs) e+06 N Testing for SLE, 13rd Itzykson Conference PUZZLES OF GROWTH, June 9-11, Saclay, France p.10/36

11 Self-avoiding walk driving process SAW with 200,000 steps; only compute driving function up to a fixed capacity T. Average number of steps in unzipped portion is ,000 samples, i.e., SAW s Plot variance of driving process, compare slope to 8/3: ± Testing for SLE, 13rd Itzykson Conference PUZZLES OF GROWTH, June 9-11, Saclay, France p.11/36

12 Self-avoiding walk driving process -cont Distribution of driving function at capacity T "density.plt" "normal.plt" Testing for SLE, 13rd Itzykson Conference PUZZLES OF GROWTH, June 9-11, Saclay, France p.12/36

13 Distorted models Take loop-erased random walk (LERW), self-avoiding walk (SAW), and percolation exploration process. LERW and percolation are proven to converge to SLE 2 and SLE 6. SAW is conjectured to converge to SLE 8/3. Distort them: (x, y) (x, λy) Easy to see this is not another SLE. No simple relation between the driving function for γ and distorted γ. Testing for SLE, 13rd Itzykson Conference PUZZLES OF GROWTH, June 9-11, Saclay, France p.13/36

14 Distorted models - cont. For each model, we compute 100,000 samples of γ and distort the curves, λ = 0.9, 0.95, 1., 1.05, 1.1. Then we compute the driving function of the distorted curve up to a fixed capacity that corresponds to about 10,000 steps. Result: collection of samples of the driving process of the distorted model. Do various statistical tests to see if this process is a Brownian motion. Invariance under (x, y) ( x, y) implies E[U t ] = 0. First plot variance E[U 2 t ] vs. t. Scaling E[U 2 t ] = ct, even if scaling limit is not an SLE. With no distortion: LERW: c = ± SAW: c = ± percolation: c = ± Testing for SLE, 13rd Itzykson Conference PUZZLES OF GROWTH, June 9-11, Saclay, France p.14/36

15 Distribution at fixed time Points are a histogram for the density of U T / T for the LERW, SAW and percolation with distortion λ = Curves are normal densities, variance from a least square fit of distorted LERW distorted SAW distorted percolation Testing for SLE, 13rd Itzykson Conference PUZZLES OF GROWTH, June 9-11, Saclay, France p.15/36

16 Normality at fixed time First statistical test: fix t, is distribution of U t normal? Kolmogorov-Smirnov test: Let F be cumulative distribution of U t. Then F(U t ) is uniformly distributed on [0, 1]. Let X 1, X 2,,X N be N samples of U t, and let X (1) < X (2) < X (N) be these numbers arranged in increasing order. D = max 1 k N F(X (k)) k N + 1 2N Test null hypothesis that U t has the distribution F. For example, P( ND > 1.36) is approximately 5%. We perform this Kolmogorov-Smirnov test for two values of the time, T and T/2. Testing for SLE, 13rd Itzykson Conference PUZZLES OF GROWTH, June 9-11, Saclay, France p.16/36

17 Kolmogorov-Smirnov results - SAW Table contains p-values λ N D(T/2) D(T) 5, , , , , , , , , , Testing for SLE, 13rd Itzykson Conference PUZZLES OF GROWTH, June 9-11, Saclay, France p.17/36

18 Independence of increments Let 0 < t 1 < t 2 < < t n = T, X j = U tj U tj 1. For Brownian motion, X j are independent; each is normal with mean zero and variance κ(t j t j 1 ). Test this joint distribution with a χ 2 goodness-of-fit test. Divide the possible values of (X 1, X 2,,X n ) into m cells, Count number of samples in each cell, O j. Brownian motion expected number is E j. χ 2 = m j=1 (O j E j ) 2 E j Testing for SLE, 13rd Itzykson Conference PUZZLES OF GROWTH, June 9-11, Saclay, France p.18/36

19 Independence of increments - cont Three choices of cells a. n = 10, use only signs of the X j. So 2 10 = 1024 cells. b. n = 5, use only signs of the X j. So 2 5 = 32 cells. c. n = 2. Look at quartile of X 1 and X 2. So 16 cells. First two statistics do not involve κ. Testing for SLE, 13rd Itzykson Conference PUZZLES OF GROWTH, June 9-11, Saclay, France p.19/36

20 Independence of increments - SAW λ N χ 2 a χ 2 b χ 2 c 5, , , , , , , , , , Testing for SLE, 13rd Itzykson Conference PUZZLES OF GROWTH, June 9-11, Saclay, France p.20/36

21 Gradient percolation We consider another model that is not SLE that we refer to as gradient-gradient percolation. For off critical percolation on a unit lattice, the length scale is Gradient percolation we can take L = (p 1/2) 4/3 p = cx The choice of c sets a length scale. Exploration process will live in a vertical strip. Driving process will be stationary for large t. Testing for SLE, 13rd Itzykson Conference PUZZLES OF GROWTH, June 9-11, Saclay, France p.21/36

22 Gradient gradient percolation L = (p 1/2) 4/3 Gradient-gradient percolation: p = cy 7/4 x At height y, width of the path is given by setting the length scale for p equal to x: x = [y 7/4 x] 4/3 = y 7/3 x 4/3 So y = x, i.e., the horizontal width is of the same order as the height. Scaling limit should be invariant under dilations Driving function variance linear in t. Testing for SLE, 13rd Itzykson Conference PUZZLES OF GROWTH, June 9-11, Saclay, France p.22/36

23 Gradient gradient percolation - cont. Sample with 50,000 steps, c=0.01 Testing for SLE, 13rd Itzykson Conference PUZZLES OF GROWTH, June 9-11, Saclay, France p.23/36

24 Gradient gradient percolation - variance 100,000 samples with 20,000 steps, c = 0.01 Variance vs. t. Slope (5.0488) estimates κ. 0.6 Variance of driving function vs t, c=0.01, 20K steps "driving_var.plt" *x Variance t Testing for SLE, 13rd Itzykson Conference PUZZLES OF GROWTH, June 9-11, Saclay, France p.24/36

25 Gradient gradient percolation - normality Distribution of U t at fixed t vs. normal. Kolmogov-Smirnov: nd = , p value is Grad-grad percolation: distrib of driving function at fixed t; c=0.01, 20K steps "density.plt" "normal.plt" Testing for SLE, 13rd Itzykson Conference PUZZLES OF GROWTH, June 9-11, Saclay, France p.25/36

26 Gradient gradient percolation - independence χ 2 tests for independence of increments: 10 increments: χ 2 a = with 1024 df Z = p value= increments: χ 2 b = with 32 df Z = , p value = increments, χ 2 c = with 16 df Z = , p value = Testing for SLE, 13rd Itzykson Conference PUZZLES OF GROWTH, June 9-11, Saclay, France p.26/36

27 Unzipping - practical issues Difference between using vertical slits vs. tilted slits for the elementary conformal maps h j is small. Given that the vertical slit map is considerably faster and easier to implement, we see no reason to use the tilted slit map. We have also shown that the speed of this algorithm can be increased dramatically using power series approximations of certain analytic functions. The loss of accuracy from this seriers approximation is extremely small, insignificant compared to the effect of changing the number of points used to define the curve we are unzipping or compared to the difference between using vertical slits or tilted slits. Testing for SLE, 13rd Itzykson Conference PUZZLES OF GROWTH, June 9-11, Saclay, France p.27/36

28 Conclusions/Future work Conclusions: Hard to distinguish something close to SLE from SLE. SAW with 5% distortion: need 100,000 samples. Best statistical test for Brownian motion depends on model. Future work: Compare testing for SLE using driving process with other tests Can you learn anything about driving process of off-critical models? Adaptive computation of driving process Testing for SLE, 13rd Itzykson Conference PUZZLES OF GROWTH, June 9-11, Saclay, France p.28/36

29 Need for adaptive computation Driving process of 20,000 step percolation exploration process Testing for SLE, 13rd Itzykson Conference PUZZLES OF GROWTH, June 9-11, Saclay, France p.29/36

30 κ s Model λ κ LERW ± LERW ± SAW ± SAW ± percolation ± percolation ± Testing for SLE, 13rd Itzykson Conference PUZZLES OF GROWTH, June 9-11, Saclay, France p.30/36

31 Kolmogorov-Smirnov results - LERW λ N D(T/2) D(T) 5, , , , , , , , , , Testing for SLE, 13rd Itzykson Conference PUZZLES OF GROWTH, June 9-11, Saclay, France p.31/36

32 Independence of increments - LERW λ N χ 2 a χ 2 b χ 2 c 5, , , , , , , , , , Testing for SLE, 13rd Itzykson Conference PUZZLES OF GROWTH, June 9-11, Saclay, France p.32/36

33 Kolmogorov-Smirnov results - SAW λ N D(T/2) D(T) 5, , , , , , , , , , Testing for SLE, 13rd Itzykson Conference PUZZLES OF GROWTH, June 9-11, Saclay, France p.33/36

34 Independence of increments - SAW λ N χ 2 a χ 2 b χ 2 c 5, , , , , , , , , , Testing for SLE, 13rd Itzykson Conference PUZZLES OF GROWTH, June 9-11, Saclay, France p.34/36

35 Kolmogorov-Smirnov results - percolation λ N D(T/2) D(T) 5, , , , , , , , , , Testing for SLE, 13rd Itzykson Conference PUZZLES OF GROWTH, June 9-11, Saclay, France p.35/36

36 Independence of increments - percolation λ N χ 2 a χ 2 b χ 2 c 5, , , , , , , , , , Testing for SLE, 13rd Itzykson Conference PUZZLES OF GROWTH, June 9-11, Saclay, France p.36/36