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Fractal Properties of the Schramm-LoewnerEvolution (SLE)
Gregory F. Lawler
Department of MathematicsUniversity of Chicago5734 S. University Ave.
Chicago, IL 60637
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!