Fractal Properties of the Schramm-LoewnerEvolution (SLE)

Gregory F. Lawler

Department of MathematicsUniversity of Chicago5734 S. University Ave.

Chicago, IL 60637

lawler@math.uchicago.edu

December 12, 2012

OUTLINE OF TALK

I The Schramm-Loewner evolution (SLEκ) is a family ofrandom fractal curves that arise as limits of models instatistical physics.

I One reason that they are interesting is that they giveexamples of nontrivial curves for which one can prove factsabout the fractal and multifractal structure.

I In this talk I will give an introduction to the curves, startingwith some discrete models and then giving the definition.

I Then I will discuss recent rigorous work on the curvesthemselves. Here we will concentrate on SLE and not on thediscrete processes.

SELF-AVOIDING WALK (SAW)

I Model for polymer chains — polymers are formed bymonomers that are attached randomly except for aself-avoidance constraint.

ω = [ω0, . . . , ωn], ωj ∈ Z2, |ω| = n

|ωj − ωj−1| = 1, j = 1, . . . , n

ωj 6= ωk , 0 ≤ j < k ≤ n.

I Critical exponent ν: a typical SAW has diameter about |ω|ν .

I If no self-avoidance constraint ν = 1/2; for 2-d SAW Florypredicted ν = 3/4.

0 N

N

z w

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-dimensionalβ large — typical path is one-dimensionalβc — typical path is (1/ν)-dimensional

Choose β = βc ; 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 measureon curves connecting boundary points of the square.

z w

0 N

N

z w

Similarly, if we fix D ⊂ C, we can consider walks restricted to thedomain D

z w

Predict that these probability measures are conformally invariant.

SIMPLE RANDOM WALK

0 N

N

z w

I Simple random walk — no self-avoidance constraint.Criticality: each walk ω gets weight (1/4)|ω|.

I Scaling limit is Brownian motion which is conformallyinvariant (Levy).

LOOP-ERASED RANDOM WALK

Start with simple random walks and erase loops in chronologicalorder to get a path with no self-intersections.

Limit should be a measure on paths with no self-intersections.

z w

ASSUMPTIONS ON SCALING LIMIT

Probability measure µ#D (z ,w) on curves connecting boundarypoints of a domain D.

f

f(w) f(z)

z w

I 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) (Riemannmapping theorem).

What is meant by the image f ◦ γ of a curve γ : [0,T ]→ C?

I One possibility is to consider curves modulo reparametrizationso that we do not care how “fast” we traverse f ◦ γ.

I If the curve γ has fractal dimension d , then the “natural”parametrization transforms as a d-dimensional measure. Thatis, the time to traverse f ◦ γ[r , s] is∫ s

r|f ′(γ(t))|d dt.

I For Brownian motion, the fractal dimension of the paths isd = 2 and Levy’s result uses that change the parametrization.

I We first consider paths modulo reparametrization and laterdiscuss the correct parametrization.

I Domain Markov property Given γ[0, t], the conditionaldistribution on γ[t,∞) is the same as

µH\γ(0,t](γ(t),∞).

γ (t)

I Satisfied on discrete level by SAW and LERW, but not bysimple random walk.

LOEWNER EQUATION IN UPPER HALF PLANE

I Let γ : (0,∞)→ H be a simple curve with γ(0+) = 0 andγ(t)→∞ as t →∞.

I gt : H \ γ(0, t]→ H

Ut

gt(t)

0

γ

I Can reparametrize (by capacity) so that

gt(z) = z +2t

z+ · · · , z →∞

I gt satisfies

∂tgt(z) =2

gt(z)− Ut, g0(z) = z .

Moreover, Ut = gt(γ(t)) is continuous.

(Schramm) Suppose γ is a random curve satisfying conformalinvariance and Domain Markov property. Then Ut must be arandom continuous curve satisfying

I For every s < t, Ut − Us is independent of Ur , 0 ≤ r ≤ s andhas the same distribution as Ut−s .

I c−1 Uc2t has the same distribution as Ut .

Therefore, Ut =√κBt where Bt is a standard (one-dimensional)

Brownian motion.

The (chordal) Schramm-Loewner evolution with parameter κ(SLEκ) is the solution obtained by choosing Ut =

√κBt .

(Rohde-Schramm) Solving the Loewner equation with a Brownianinput gives a random curve.

The qualitative behavior of the curves varies greatly with κ

I 0 < κ ≤ 4 — simple (non self intersecting) curve

I 4 < κ < 8 — self-intersections (but not crossing); notplane-filling

I 8 ≤ κ <∞ — plane-filling

(Beffara) For κ < 8, the Hausdorff dimension of the paths is

1 +κ

8.

NATURAL PARAMETRIZATION/LENGTH

I Start with pathω = [ω0, ω1, . . .]

in Z2. Assume it has “fractal dimension” d .

I Letγ(n)(t) = n−1 ωnd t .

Hope to take limit as n→∞.

I For simple random walk, d = 2 and γ(t) is Brownian motion(Donsker’s theorem)

I Expect similar result for SAW (d = 4/3, SLE8/3) and LERW(d = 5/4, SLE2).

SCALING RULE

I Suppose γ has “natural parametrization”.

I If f : D → f (D) is a conformal transformation, then the timeneeded to traverse f (γ[0, t]) is∫ t

0|f ′(γ(s))|d ds.

I While SAW and LERW have limits that are SLEκ, thecapacity parametrization does not have this property.

I In fact, the capacity parametrization is singular with respectto the natural length.

I Problem: can we define the natural length for SLEκ?

Ut

gt(t)

0

γ

ft(z) = g−1t (z + Ut)

I In capacity para., time to traverse gt(γ[t, t + ∆t]) is ∆t.

I This should not be true for natural length.

I For natural length, need to understand g ′t(w) near γ(t) orf ′t (z) near zero.

GREEN’S FUNCTION

I The SLE Green’s function (for chordal SLE from w1 to w2 inD) is defined by

GD(z ;w1,w2) = limε↓0

εd−2 P{dist(z , γ) ≤ ε}.

Defined up to multiplicative constant.

I This was computed by Rohde-Schramm and L. first showedthe limit exists with distance replaced by conformal radius.More recently, L-Rezaei have proved the limit above exists.

I Let G (z) = GH(z ; 0,∞). Then

G (z) = [Im z ]d−2 [sin arg z ]8κ−1.

RIGOROUS DEFINITION

Let γ be SLEκ in H parametrized by capacity. γt = γ[0, t].

I Let Θt be the natural length of γt spent in a bounded domainD.

I Heuristic:

E[Θ∞] =

∫DG (z) dA(z).

I

E[Θ∞ | γt ] = Θt + Ψt ,

Ψt =

∫DGH\γt (z ; γ(t),∞) dA(z).

I Θt is the increasing process that makes Ψt + Θt a martingale.(Doob-Meyer decomposition)

I (L-Sheffield) The natural length is well defined forκ < 5.021 · · · . It is Holder continuous.

I (L-Zhou) Exists for all κ < 8. This proof relies on a slightgeneralization of a hard estimate of Beffara. It also uses atwo-point Green’s function (L-Werness). An improved versionusing a two-point time-dependent Green’s function has beengiven (L-Rezaei)

I (L-Rezaei) The natural parametrization is given by thed-dimensional Minkowski content

Θt = c limε↓0

εd−2Area{z : dist(z , γ[0, t]) < ε}.

I (Rezaei) The d-dimensional Hausdorff measure of the path iszero.

TIP MULTIFRACTAL SPECTRUM(work with F. Johansson Viklund)

I Study behavior of |g ′t | near γ(t) or |f ′t | near 0.

Ut

gt(t)

0

γ

I Let Λβ denote the set of t such that as y ↓ 0,

|f ′t (iy)| ≈ y−β.

I Closely related to behavior of harmonic measure near the tipof the curve.

I Let

ρ = ρκ(β) =κ

8(β + 1)

[(κ+ 4

κ

)(β + 1)− 1)

]2.

I (L-Johansson Viklund) With probability one, if ρ < 2,

dimh(Λβ) =2− ρ

2

dimh(γ(Λβ)) =2dimh(Λβ)

1− β=

2− ρ1− β

.

I If ρ > 2, then Λβ = ∅.

I dimh(Λβ) depends on the capacity parametrization of thepath. The quantity dimh(γ(Λβ)) is independent of theparametrization.

I Finding the formula for ρ requires analyzing E[|f ′t (i)|λ

]for

large t.I Computing the almost sure multifractal spectrum requires

more work than just computing ρ. There are tricky secondmoment estimates involved.

I Nonrigorous (“physicist”) treatments of multifractal spectrummay compute ρ but do not do the second moment workneeded to make this an almost sure statement.

I ρ can be computed by analyzing E[|f ′t (i)|λ

]for large t. For

r < 2a + 12 , (a = 2/κ)

r(λ) = 2a + 1−√

(2a + 1)2 − 4aλ,

ζ(λ) = λ− r

2a

−β(λ) = ζ ′(λ) = 1− 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)| ≈ tβ(λ)/2,

P{|f ′t2(i)| ≈ tβ/2} ≈ t−ρ, ρ = λβ + ζ.

I The technique is to find an appropriate martingale and useGirsanov theorem to analyze the paths in the measure tiltedby the martingale.

I As an example, consider the natural parametrization. Thiscorresponds to λ = d = 1 + κ

8 .

r = 1, λ = d , ζ = 2− d ,β

2= d − 3

2=

1

4a− 1

2

E[|f ′1(i/

√n)|d

]= E

[|f ′n(i)|d

]� n

d2−1

P{|f ′1(i/√n)| ≈ nd−

32 } ≈ n−(d

2−2d+1)

I The Hausdorff dimension of the set of times t ∈ [0, 1] with

|f ′t (i/√n)| ≈ nd−

32 equals

1− (d2 − 2d + 1) = d(2− d) ∈ (0, 1).

I The natural parametrization is carried on a set of ?? ofdimension d(2− d).

I The dimension of points γ(t) satisfying this is d

HOLDER CONTINUITY OF γ

I Consider γ(t), ε ≤ t ≤ 1 (with capacity parametrization)I γ(t), ε ≤ t ≤ 1 is Holder continuous of order α < α∗ but notα > α∗ where

α∗ = 1− κ

24 + 2κ− 8√

8 + κ.

I One direction shown by Joan Lind. Other direction byL-Johansson Viklund.

I α∗ > 0 unless κ = 8. Showing that the curve exists is muchharder for κ = 8 than other values.

SOME OPEN PROBLEMS

I Show that for κ < 8, SLE with the natural parametrization isHolder continuous for α < 1/d .

I Find modulus of continuity for SLE8 in capacityparametrization.

I Extend multifractal spectrum analysis to entire path, not justtip (the “first moment” calculations have been done but notthe second moment analysis for almost sure behavior).

I Show that discrete processes converge to SLE in the naturalparametrization. Work is being done on the loop-erased walk.

I Find a Hausdorff gauge function for which the Hausdorffmeasure of SLE paths is finite and positive.

THANK YOU!