PATH PROPERTIES OF SCHRAMM-LOEWNER EVOLUTION (SLE) In memory of - PowerPoint PPT Presentation

PATH PROPERTIES OF SCHRAMM-LOEWNER EVOLUTION (SLE) In memory of Oded Schramm Gregory F. Lawler Department of Mathematics University of Chicago 5734 S. University Ave. Chicago, IL 60637 lawler@math.uchicago.edu August, 2009 Gregory F. Lawler PATH PROPERTIES OF SCHRAMM-LOEWNER EVOLUTION (

Loewner equation a = 2 a ∂ t g t ( z ) = , g 0 ( z ) = z , κ > 0 , g t ( z ) − U t U t = − B t standard Brownian motion . Since U : [0 , ∞ ) → R is continuous, there exists simp. conn. H t ⊂ H such that g t maps H t conformally onto H with � 1 g t ( z ) = z + a � z + O , z → ∞ . | z | 2 Does there exist a curve γ : [0 , ∞ ) → H such that H t is the unbounded component of H \ γ (0 , t ]? Is the curve simple? Is γ (0 , ∞ ) ⊂ H ? What is the Hausdorff dimension of γ (0 , t ]? Gregory F. Lawler PATH PROPERTIES OF SCHRAMM-LOEWNER EVOLUTION (

S. Rohde and O. Scrhamm (2005) Basic properties of SLE, Annals of Math. EXISTENCE OF THE PATH f t = g − 1 ˆ , f t ( z ) = f t ( z + U t ) . t Roughly speaking γ ( t ) = f t ( U t ) = ˆ f t (0). � i � � � U t + i = ˆ γ n ( t ) = f t f t , n n γ ( t ) = lim n →∞ γ n ( t ) . Goal: Show limit exists and give bounds on | γ ( s ) − γ ( t ) | . Gregory F. Lawler PATH PROPERTIES OF SCHRAMM-LOEWNER EVOLUTION (

� 1 / n | ˆ f ′ | γ ( t ) − γ n ( t ) | ≤ t ( iy ) | dy . 0 t ( iy ) | ≤ c y δ − 1 for some δ > 0, then If one can show that | ˆ f ′ | γ ( t ) − γ n ( t ) | ≤ O ( n − δ ). Given the modulus of continuity of Brownian motion, U t − U s ≈ | t − s | 1 / 2 and distortion estimates for conformal maps, it suffices (up to logarithmic factors) to show that | ˆ f ′ k / n 2 ( i / n ) | ≤ c n 1 − δ , k = 1 , 2 , . . . , n 2 Gregory F. Lawler PATH PROPERTIES OF SCHRAMM-LOEWNER EVOLUTION (

Let ǫ = 1 / n and s ≤ t ≤ s + ǫ 2 . | ˆ f ′ s ( i ǫ ) − ˆ f ′ t ( i ǫ ) | = | f ′ s ( i ǫ + U s ) − f ′ t ( i ǫ + U t ) | ≤ | f ′ s ( i ǫ + U s ) − f ′ t ( i ǫ + U s ) | + | f ′ t ( i ǫ + U s ) − f ′ t ( i ǫ + U t ) | From the Loewner equation for f we get | f ′ s ( i ǫ + U s ) − f ′ t ( i ǫ + U s ) | ≤ c | f ′ | s − t | ≤ ǫ 2 s ( i ǫ + U s ) | , Since | U t − U s | ≈ ǫ , distortion estimates give | f ′ t ( i ǫ + U s ) − f ′ t ( i ǫ + U t ) | ≤ c | f ′ t ( i ǫ + U s ) | . Hence | ˆ s ( i ǫ ) | ≈ | ˆ f ′ f ′ t ( i ǫ ) | . Gregory F. Lawler PATH PROPERTIES OF SCHRAMM-LOEWNER EVOLUTION (

Let ǫ = 1 / n and s ≤ t ≤ s + ǫ 2 . | γ ( s ) − γ ( t ) | ≤ | γ ( s ) − γ n ( s ) | + | γ n ( s ) − γ n ( t ) | + | γ n ( t ) − γ ( t ) | | γ n ( s ) − γ n ( t ) | = | f s ( U s + ǫ i ) − f t ( U t + ǫ i ) | ≤ | f s ( U s + ǫ i ) − f t ( U s + ǫ i ) | + | f t ( U s + ǫ i ) − f t ( U t + ǫ i ) | | γ n ( s ) − γ n ( t ) | can be estimated using mod of cont and disortion estimate. Boils down to how well we can estimate | ˆ f ′ k / n 2 ( i / n ) | which has the same distribution as | ˆ f ′ k ( i ) | . Gregory F. Lawler PATH PROPERTIES OF SCHRAMM-LOEWNER EVOLUTION (

ˆ f ′ t ( z ) has the same distribution as h ′ t ( z ) where h t follows the reverse Loewner flow: a ∂ t h t ( z ) = U t − h t ( z ) , h 0 ( z ) = z . Z t = Z t ( z ) = X t + iY t = h t ( z ) − U t t ( z ) | λ Y ζ t [sin arg Z t ] − r M t ( z ) = | h ′ − r 2 � � λ = ζ + r 1 + 1 2 a = 4 a . 2 a M t ( z ) is a martingale, E [ M t ( i )] = 1 . Gregory F. Lawler PATH PROPERTIES OF SCHRAMM-LOEWNER EVOLUTION (

One hopes that Y t ( i ) ≍ t 1 / 2 , [sin arg Z t ( i )] ≍ 1, in which case we can conclude � t ( i ) | λ � | h ′ ≍ t − ζ/ 2 . E This is correct if r < 2 a + 1 2 . This estimate is good enough to prove existence of curve for κ � = 8. For κ = 8 the existence follows from work of LSW of SLE κ as limit of Peano curve. (Lind, Johansson-L.) γ ( t ) , ǫ ≤ t ≤ 1 is H¨ older continuous of order α < α ∗ but not α > α ∗ where κ α ∗ = 1 − 24 + 2 κ − 8 √ 8 + κ. α ∗ > 0 unless κ = 8. Gregory F. Lawler PATH PROPERTIES OF SCHRAMM-LOEWNER EVOLUTION (

(1/2 times) CONFORMAL RADIUS Given simp. conn. D and w ∈ D define Υ D ( w ) = 1 2 f ′ (0) where f : D → D is the conformal transformation with f (0) = w , f ′ (0) > 0 . The factor 1 / 2 is a convenience so that Υ H ( i ) = 1. Υ D ( w ) ≤ dist ( w , ∂ D ) ≤ 2 Υ D ( w ) . 2 Scaling rule Υ f ( D ) ( f ( w )) = | f ′ ( w ) | Υ D ( w ) . Gregory F. Lawler PATH PROPERTIES OF SCHRAMM-LOEWNER EVOLUTION (

Given simp. conn. D , distinct boundary points z 1 , z 2 and w ∈ D define Θ D ( w ; z 1 , z 2 ) = arg F ( w ) , S D ( w ; z 1 , z 2 ) = sin arg F ( w ) where F : D → H is a conformal transformation with F ( z 1 ) = 0 , F ( z 2 ) = ∞ . Gregory F. Lawler PATH PROPERTIES OF SCHRAMM-LOEWNER EVOLUTION (

Let γ be SLE κ from 0 to ∞ in H and H t the unbounded component of H \ γ (0 , t ]. Θ t = Θ H t ( w ; γ ( t ) , ∞ ) . If we reparametrize time (˜ Θ t = Θ σ ( t ) ) so that log Υ t decays linearly, the Loewner equation gives d ˜ Θ t = (1 − 2 a ) cot ˜ Θ t dt + dB t . If a ≤ 1 / 4 ( κ ≥ 8), the process never hits zero which implies that the conformal radius goes to zero, i.e., SLE κ hits points. For κ < 8, SLE does not hit points. For κ < 8 can determine whether curve goes to right or left of w . Gregory F. Lawler PATH PROPERTIES OF SCHRAMM-LOEWNER EVOLUTION (

FRACTAL DIMENSION OF γ (0 , ∞ ) FOR κ < 8 Let d be fractal dimension Standard heuristic argument indicates that P { dist [ w , γ (0 , ∞ )] ≤ ǫ } ≈ ǫ 2 − d . Similarly we could write P { Υ ∞ ≤ ǫ } ≈ ǫ 2 − d . Gregory F. Lawler PATH PROPERTIES OF SCHRAMM-LOEWNER EVOLUTION (

Assume there is a function G D ( w ; z 1 , z 2 ) such that if γ is SLE κ from z 1 to z 2 in D , P { Υ ≤ ǫ } ∼ G D ( w ; z 1 , z 2 ) ǫ 2 − d , ǫ → 0 , where Υ = Υ D \ γ ( w ) . Conformal invariance of SLE implies the scaling rule G D ( w ; z 1 , z 2 ) = | F ′ ( w ) | 2 − d G F ( D ) ( F ( w ); F ( z 1 ) , F ( z 2 )) . Also, M t = G D \ γ (0 , t ] ( w ; γ ( t ) , z 2 ) is a local martingale. Gregory F. Lawler PATH PROPERTIES OF SCHRAMM-LOEWNER EVOLUTION (

Using Itˆ o’s formula, we can find that d = 1 + κ 8 and (up to a multiplicative constant) β = 8 G D ( w ; z 1 , z 2 ) = Υ D ( w ) d − 2 S D ( w ; z 1 , z 2 ) β , κ − 1 > 0 . Gregory F. Lawler PATH PROPERTIES OF SCHRAMM-LOEWNER EVOLUTION (

Consider i ∈ H and let Υ t = Υ H t ( i ) so that Υ t ≍ dist ( i , γ (0 , t ]). M t = Υ d − 2 S t , S t = S H t ( i ; γ ( t ) , ∞ ) . t τ ǫ = inf { t : Υ t = ǫ } . If τ ǫ = ∞ , M ∞ = 0. Would like to say 1 = E [ M τ ǫ ; τ ǫ < ∞ ] = ǫ d − 2 P { τ ǫ < ∞} E [ S τ ǫ | τ ǫ < ∞ ] E [ S τ ǫ | τ ǫ < ∞ ] → c − 1 P { τ ǫ < ∞} ∼ c ∗ ǫ 2 − d . ∗ , This can be done (computing c ∗ ) using Girsanov, considering the SDE for S τ ǫ (function of ǫ ) when weighted by the martingale. Gregory F. Lawler PATH PROPERTIES OF SCHRAMM-LOEWNER EVOLUTION (

It is harder to get second moment estimates P { Υ( z ) ≤ ǫ, Υ( w ) ≤ ǫ } ≍ ǫ 2 − d ǫ 2 − d | z − w | d − 2 , which are needed to prove Hausdorff dimension rigorously. This was done by Beffara. An alternative proof can be given using the reverse Loewner flow for which the second moment estimates seem to be somewhat easier (but still take work). Gregory F. Lawler PATH PROPERTIES OF SCHRAMM-LOEWNER EVOLUTION (

NATURAL PARAMETRIZATION (L-Sheffield, L-Z. Wang) Parametrize SLE κ using the “natural” or “fractal” parameterizaton, a d -dimensional param. D — bounded domain. γ ( t ) - SLE κ in D from z 1 to z 2 defined with some param. (e.g., hcap in H ). D t component of D \ γ [0 , t ] containing z 2 on boundary. Θ t = amount of time in natural param for γ [0 , t ] . � E [Θ ∞ ] = G D ( w ; z 1 , z 2 ) dA ( w ) . D � Θ t + G D t ( w ; γ ( t ) , z 2 ) dA ( w ) D t is a martingale. Gregory F. Lawler PATH PROPERTIES OF SCHRAMM-LOEWNER EVOLUTION (