Fractal Properties of the Schramm-Loewner Evolution (SLE)

Transcription

1 Fractal Properties of the Schramm-Loewner Evolution (SLE) Gregory F. Lawler Department of Mathematics University of Chicago 5734 S. University Ave. Chicago, IL December 12, 2012

2 OUTLINE OF TALK The Schramm-Loewner evolution (SLE κ ) is a family of random fractal curves that arise as limits of models in statistical physics. One reason that they are interesting is that they give examples of nontrivial curves for which one can prove facts about the fractal and multifractal structure. In this talk I will give an introduction to the curves, starting with some discrete models and then giving the definition. Then I will discuss recent rigorous work on the curves themselves. Here we will concentrate on SLE and not on the discrete processes.

3 SELF-AVOIDING WALK (SAW) Model for polymer chains polymers are formed by monomers that are attached randomly except for a self-avoidance constraint. ω = [ω 0,..., ω n ], ω j Z 2, ω = n ω j ω j 1 = 1, j = 1,..., n ω j ω k, 0 j < k n. Critical exponent ν: a typical SAW has diameter about ω ν. If no self-avoidance constraint ν = 1/2; for 2-d SAW Flory predicted ν = 3/4.

4 N z w 0 N Each SAW from z to w gets measure e β ω. Partition function Z = Z(N, β) = e β ω. β small typical path is two-dimensional β large typical path is one-dimensional β c typical path is (1/ν)-dimensional

5 Choose β = β c ; let N. Expect Z(N, β) C(D; z, w) N 2b, divide by C(D; z, w) N 2b and hope to get a probability measure on curves connecting boundary points of the square. z w

6 N z w 0 N Similarly, if we fix D C, we can consider walks restricted to the domain D z w Predict that these probability measures are conformally invariant.

7 SIMPLE RANDOM WALK N z w 0 N Simple random walk no self-avoidance constraint. Criticality: each walk ω gets weight (1/4) ω. Scaling limit is Brownian motion which is conformally invariant (Lévy).

8 LOOP-ERASED RANDOM WALK Start with simple random walks and erase loops in chronological order to get a path with no self-intersections. Limit should be a measure on paths with no self-intersections. z w

9 ASSUMPTIONS ON SCALING LIMIT Probability measure µ # D (z, w) on curves connecting boundary points of a domain D. z w f f(z) f(w) Conformal invariance: If f is a conformal transformation f µ # D (z, w) = µ# f (D)(f (z), f (w)). For simply connected D, µ # H (0, ) determines µ# D (z, w) (Riemann mapping theorem).

10 What is meant by the image f γ of a curve γ : [0, T ] C? One possibility is to consider curves modulo reparametrization so that we do not care how fast we traverse f γ. If the curve γ has fractal dimension d, then the natural parametrization transforms as a d-dimensional measure. That is, the time to traverse f γ[r, s] is s r f (γ(t)) d dt. For Brownian motion, the fractal dimension of the paths is d = 2 and Lévy s result uses that change the parametrization. We first consider paths modulo reparametrization and later discuss the correct parametrization.

11 Domain Markov property Given γ[0, t], the conditional distribution on γ[t, ) is the same as µ H\γ(0,t] (γ(t), ). γ (t) Satisfied on discrete level by SAW and LERW, but not by simple random walk.

12 LOEWNER EQUATION IN UPPER HALF PLANE Let γ : (0, ) H be a simple curve with γ(0+) = 0 and γ(t) as t. g t : H \ γ(0, t] H γ (t) g t 0 U t Can reparametrize (by capacity) so that g t satisfies g t (z) = z + 2t z +, t g t (z) = z 2 g t (z) U t, g 0 (z) = z. Moreover, U t = g t (γ(t)) is continuous.

13 (Schramm) Suppose γ is a random curve satisfying conformal invariance and Domain Markov property. Then U t must be a random continuous curve satisfying For every s < t, U t U s is independent of U r, 0 r s and has the same distribution as U t s. c 1 U c 2 t has the same distribution as U t. Therefore, U t = κ B t where B t is a standard (one-dimensional) Brownian motion. The (chordal) Schramm-Loewner evolution with parameter κ (SLE κ ) is the solution obtained by choosing U t = κ B t.

14 (Rohde-Schramm) Solving the Loewner equation with a Brownian input gives a random curve. The qualitative behavior of the curves varies greatly with κ 0 < κ 4 simple (non self intersecting) curve 4 < κ < 8 self-intersections (but not crossing); not plane-filling 8 κ < plane-filling (Beffara) For κ < 8, the Hausdorff dimension of the paths is 1 + κ 8.

15 NATURAL PARAMETRIZATION/LENGTH Start with path ω = [ω 0, ω 1,...] in Z 2. Assume it has fractal dimension d. Let Hope to take limit as n. γ (n) (t) = n 1 ω n d t. For simple random walk, d = 2 and γ(t) is Brownian motion (Donsker s theorem) Expect similar result for SAW (d = 4/3, SLE 8/3 ) and LERW (d = 5/4, SLE 2 ).

16 SCALING RULE Suppose γ has natural parametrization. If f : D f (D) is a conformal transformation, then the time needed to traverse f (γ[0, t]) is t 0 f (γ(s)) d ds. While SAW and LERW have limits that are SLE κ, the capacity parametrization does not have this property. In fact, the capacity parametrization is singular with respect to the natural length. Problem: can we define the natural length for SLE κ?

17 γ (t) g t 0 U t f t (z) = g 1 t (z + U t ) In capacity para., time to traverse g t (γ[t, t + t]) is t. This should not be true for natural length. For natural length, need to understand g t(w) near γ(t) or f t (z) near zero.

18 GREEN S FUNCTION The SLE Green s function (for chordal SLE from w 1 to w 2 in D) is defined by G D (z; w 1, w 2 ) = lim ɛ 0 ɛ d 2 P{dist(z, γ) ɛ}. Defined up to multiplicative constant. This was computed by Rohde-Schramm and L. first showed the limit exists with distance replaced by conformal radius. More recently, L-Rezaei have proved the limit above exists. Let G(z) = G H (z; 0, ). Then G(z) = [Im z] d 2 [sin arg z] 8 κ 1.

19 RIGOROUS DEFINITION Let γ be SLE κ in H parametrized by capacity. γ t = γ[0, t]. Let Θ t be the natural length of γ t spent in a bounded domain D. Heuristic: E[Θ ] = D G(z) da(z). E[Θ γ t ] = Θ t + Ψ t, Ψ t = G H\γt (z; γ(t), ) da(z). D Θ t is the increasing process that makes Ψ t + Θ t a martingale. (Doob-Meyer decomposition)

20 (L-Sheffield) The natural length is well defined for κ < It is Hölder continuous. (L-Zhou) Exists for all κ < 8. This proof relies on a slight generalization of a hard estimate of Beffara. It also uses a two-point Green s function (L-Werness). An improved version using a two-point time-dependent Green s function has been given (L-Rezaei) (L-Rezaei) The natural parametrization is given by the d-dimensional Minkowski content Θ t = c lim ɛ 0 ɛ d 2 Area{z : dist(z, γ[0, t]) < ɛ}. (Rezaei) The d-dimensional Hausdorff measure of the path is zero.

21 TIP MULTIFRACTAL SPECTRUM (work with F. Johansson Viklund) Study behavior of g t near γ(t) or f t near 0. γ (t) g t 0 U t Let Λ β denote the set of t such that as y 0, f t (iy) y β. Closely related to behavior of harmonic measure near the tip of the curve.

22 Let ρ = ρ κ (β) = κ 8(β + 1) [( ) κ (β + 1) 1)]. κ (L-Johansson Viklund) With probability one, if ρ < 2, dim h (Λ β ) = 2 ρ 2 If ρ > 2, then Λ β =. dim h (γ(λ β )) = 2dim h(λ β ) 1 β = 2 ρ 1 β.

23 dim h (Λ β ) depends on the capacity parametrization of the path. The quantity dim h (γ(λ β )) is independent of the parametrization. Finding the formula for ρ requires analyzing E [ f t (i) λ] for large t. Computing the almost sure multifractal spectrum requires more work than just computing ρ. There are tricky second moment estimates involved. Nonrigorous ( physicist ) treatments of multifractal spectrum may compute ρ but do not do the second moment work needed to make this an almost sure statement.

24 ρ can be computed by analyzing E [ f t (i) λ] for large t. For r < 2a + 1 2, (a = 2/κ) r(λ) = 2a + 1 (2a + 1) 2 4aλ, ζ(λ) = λ r 2a β(λ) = ζ 1 (λ) = 1 (2a + 1) 2 4aλ, [ E f t (i) λ] t ζ(λ)/2 and a typical path when weighted by f t (i) λ has f t (i) t β(λ)/2, P{ f t 2 (i) t β/2 } t ρ, ρ = λ β + ζ. The technique is to find an appropriate martingale and use Girsanov theorem to analyze the paths in the measure tilted by the martingale.

25 As an example, consider the natural parametrization. This corresponds to λ = d = 1 + κ 8. β r = 1, λ = d, ζ = 2 d, 2 = d 3 2 = 1 4a 1 2 [ E ˆf 1(i/ n) d] [ = E ˆf n(i) d] n d 2 1 P{ ˆf 1(i/ n) n d 3 2 } n (d 2 2d+1) The Hausdorff dimension of the set of times t [0, 1] with ˆf t (i/ n) n d 3 2 equals 1 (d 2 2d + 1) = d(2 d) (0, 1). The natural parametrization is carried on a set of?? of dimension d(2 d). The dimension of points γ(t) satisfying this is d

26 HÖLDER CONTINUITY OF γ Consider γ(t), ɛ t 1 (with capacity parametrization) γ(t), ɛ t 1 is Hölder continuous of order α < α but not α > α where α = 1 κ κ κ. One direction shown by Joan Lind. Other direction by L-Johansson Viklund. α > 0 unless κ = 8. Showing that the curve exists is much harder for κ = 8 than other values.

27 SOME OPEN PROBLEMS Show that for κ < 8, SLE with the natural parametrization is Hölder continuous for α < 1/d. Find modulus of continuity for SLE 8 in capacity parametrization. Extend multifractal spectrum analysis to entire path, not just tip (the first moment calculations have been done but not the second moment analysis for almost sure behavior). Show that discrete processes converge to SLE in the natural parametrization. Work is being done on the loop-erased walk. Find a Hausdorff gauge function for which the Hausdorff measure of SLE paths is finite and positive.

28 THANK YOU!