Conformally Invariant Processesand the Schramm–Loewner Evolution

Laurence S. Fieldmath.uchicago.edu/�laurie

Sydney Dynamics Group 1 August 2014

Critical phenomena in statistical mechanics

We study systems on a discrete lattice that exhibit a phase change as aparameter (usually temperature) varies.

Physicists expect the following behaviour:

High temperatures exhibit short-range correlations (decayingexponentially with distance).

Low temperatures exhibit correlations at all distances.

At the critical temperature, correlations decay as a power law withdistance, leading to fractal-like behaviour, and nontrivial criticalexponents.

The critical temperature depends on the exact lattice shape.

Universality says that the critical exponents do not depend on thelattice shape, apart from the dimension of the lattice.

Square-lattice Ising model

On a domain D � Z2, a spin configuration � W D ! f1;�1g hasenergy (Hamiltonian)

H.�/ D �Xx�y

�.x/ �.y/:

The usual parameter is inverse temperature ˇ D 1=kBT .

The probability of a spin configuration � is given by the Gibbsmeasure

P.�/ De�ˇH.�/

Z; where Z D

X�

e�ˇH.�/

is called the partition function.

The critical inverse temperature is ˇc D .1Cp2/=2 � 0:440687.

Square-lattice Ising model

Critical, ˇ D 0:440687 Subcritical, ˇ D 0:45

Simulations due to Vincent Beffara

Percolation

Colour each

(vertex (site percolation)edge (bond percolation)

independently black or white with probability p or 1 � p.

Critical bond percolation cluster,square lattice, p D 1

2

Critical site percolationwith exploration process,triangular lattice, p D 1

2

Conformal invariance in two dimensions

Belavin, Polyakov and Zamolodchikov (1984) predicted thattwo-dimensional critical systems display conformal invariance in thescaling limit, and deduced many algebraic properties of such limits.This was the beginning of conformal field theory.

Conformal invariance is a natural conjecture, because a power law forcorrelations suggests that the system looks similar at all scales.

If there is sufficient “locality” for the scale to vary from point to point,then the system should be invariant under conformal transformations.

Despite much work on conformal field theory, actually proving theconformal invariance of scaling limits has proven very difficult.

Stas Smirnov proved the conformal invariance of critical sitepercolation on the triangular lattice (2001) and the square-lattice Isingmodel (2010). He won the Fields medal for this work.

Conformal invariance on other lattices remains an open problem.

Self-avoiding walk

Consider all self-avoiding nearest-neighbour walks of length N on afixed lattice.

By submultiplicativity, the number of such walks is asymptotic to cN

for some number c > 0 called the connective constant.

Give a self-avoiding walk ! energy H.!/ D j!j (the number of stepsin !). Applying the Gibbs measure introduced earlier, the criticalinverse temperature is ˇc D log c.

Conformal invariance of self-avoiding walk is a long-standingconjecture supported by good numerical evidence [Tom Kennedy,Tony Guttmann].

Smirnov and Duminil-Copin managed to prove that the connectiveconstant of the honeycomb lattice is

p2Cp2. Despite this, the

proof of conformal invariance is probably very far off.

Self-avoiding walk

Simulations due to Nathan Clisby

Simple random walk

At each step, go to one of the neighbouringvertices with probability 1=4.

The scaling limit of simple random walkis Brownian motion [Donsker], which isconformally invariant [Paul Levy].

The outer boundary is also conformally in-variant.

Mandelbrot said he could see by visual in-spection that the dimension of the outerboundary of Brownian motion was 4=3.

Mandelbrot’s conjecture was proved by Lawler, Schramm and Wernerin 2000 using properties of the Schramm–Loewner evolution SLE6,which has the same outer boundary as Brownian motion. They alsofound the critical non-intersection exponents for Brownian motion.

A simple random walk loop with 4 million steps

A simple random walk loop with 100 million steps

Loop-erased random walk (LERW)

Run a simple random walk, butdelete every loop as soon as youfinish it.

The conformal invariance ofBrownian motion suggests thatthe scaling limit of LERW shouldbe conformally invariant. It is,though the proof is not easy.

Uniform spanning tree (UST)Pick a spanning tree of a finitelattice uniformly at random.

Amazingly, this can be doneby running loop-erased randomwalks from all vertices in anyorder, each stopped when it hitsthe existing walks [Wilson].

We can traverse a UST just aswe traverse a maze with ourright hand.

Lawler, Schramm and Wernerproved that the scaling limits ofLERW and UST are the con-formally invariant curves SLE2and SLE8 respectively.

Simulation due to Eric Farmer

Schramm–Loewner evolution (SLE)

Chordal SLE� is a conformally invariant family of random curves insimply connected planar domains D connecting marked boundarypoints a; b.

These random curves appear as the scaling limits of interfaces incritical statistical mechanics models.

Domain Markov Property: Given an initial segment t of SLE�in D, the remainder of the curve is SLE� in the slit domain D n t .

Oded Schramm’s argumentAny conformally invariant family of random curves �D;a;b thatsatisfies the domain Markov property must be SLE� for some �.

By the Riemann mapping theorem, we may work just with �H;0;1.

Consider the unique conformal map gt that sends H n t to H withgt .´/ � ´! 0 as ´!1. We may parametrize the curve so thatgt .´/ D ´C 2t=´CO.´

�2/ as ´!1.

The tip .t/ is sent to some real point Ut , which is called the drivingfunction for reasons that will become apparent.

By the domain Markov property, .t;1/ has law �Hn t ; .t/;1.

By conformal invariance, the image of .t;1/ under gt has law�H;Ut ;1, which is a translate of �H;0;1.

Therefore, the process Ut has independent, stationary increments.

One can show that Ur2t agrees in law with rUt , so the drivingfunction is Brownian motion with some variance parameter � > 0.

The definition of SLE

The Loewner equation from conformal mapping theory describes howa domain slit by a curve evolves in terms of its driving function.

Using Brownian motion as the driving function in the Loewnerequation, we get the definition of SLE:

SLE� in H from 0 to1 is defined to be the random curve t suchthat, if Ht is the unbounded component of H n t , then

@

@tgt .´/ D

2

gt .´/ � Ut; g0.´/ D ´; ´ 2 H;

where Ut is a driftless Brownian motion with variance parameter �.

It is not easy to show that there is such a curve. In the case � D 8 it isknown only as a corollary of the convergence of UST.

Analysing SLE using stochastic calculusFrom the definition of SLE, simple stochastic differential equationsarise which tell us a lot about the curve’s properties.

For instance, if x > 0, by setting Xt D gt .x/ � Ut we see that

dXt D2

Xtdt � dUt

which is a Bessel process of dimension 4=� C 1.

Using Ito’s formula, it is easy to see that X1�4=�t (or logXt if � D 4)is a local martingale. We can deduce that, with probability one, theprocess Xt hits 0 if and only if � � 4.

With more work, we obtain the phases of SLE:

I If � � 4, SLE� curves are simple.

I If 4 < � < 8, SLE� curves touch themselves and the boundary.

I If � � 8, SLE� curves cover every point of the domain.

SLE and the Gaussian Free FieldIn works of Sheffield and Miller, a new theory called “ImaginaryGeometry” has been developed.

The idea is that SLE curves can be obtained as flow lines of the vectorfield eih=�, where h is a random generalized function called theGaussian free field (GFF) and � D �.�/ is a constant. The theory isbased on a coupling between SLE and the GFF due to Dubedat.

The fact that h is not a true function accounts for the fact that the SLEcurves are not smooth.

Recently, Brent Werness has developed a fast simulation algorithm forSLE using the ideas of Imaginary Geometry.

By simulating the free field only at the desired spatial resolution, heachieves a cost of O.logN/ per point, compared to O.N/ for thenaıve algorithm, or O.N ˛/, ˛ � 0:4 in work of Tom Kennedy.

The simulations are much less prone to the inaccuracy that comesfrom the parametrization by capacity at infinity.

Werness’ simulation algorithm

Let � be a measure on self-avoiding nearest-neighbor paths in adiscrete lattice. Restricting � to the paths that pass through a markedvertex ´, we obtain a pinned measure �´.

Let ´ run over all vertices in the lattice, and form the aggregate� D

P´ �´

of all the pinned measures.

The aggregate compared with the original

In this discrete aggregate �, each path is counted once for everyvertex that it traverses, each time with the original mass �. /.Therefore,

�. / D j j�. /:

In other words, the aggregate � is the initial measure � biased by thepath length j j.

Other discrete models

The general procedure outlined above is:

I Aggregate pinned or rooted measures;

I Forget the roots;

I Remove the biasing factor of path length, and thus

I Recover a natural unpinned or unrooted measure.

The same procedure is used:

I in Markov chain loop measures,e.g. the random walk loop measure[Lawler and Trujillo Ferreras];

I in the conformally invariant Brownian loop measure[Lawler and Werner].

Pinning and length-biasing for SLE

It is natural to ask whether the relation between pinned measures andlength-biasing holds in the continuum for SLE curves.

To understand this, we must first answer two questions:

1. How do we construct the “pinned measure” of all SLE curvesthat pass through a marked point?

2. How do we measure the length of an SLE curve?

The SLE� Green’s functionLet � < 8. If is a chordal SLE� curve in a simply connecteddomain D between marked boundary points, the SLE� Green’sfunction GD is the normalized probability of passing through a givenpoint. To be precise,

Pfdist.´; / < �g � �2�d GD.´/ as � ! 0C; for ´ 2 D;

where d D 1C �=8 is the Hausdorff dimension of SLE� curves.

The Green’s function is characterized by two properties that areintuitively clear from its definition as a normalized probability:

I G is covariant under conformal maps f :

GD.´/ D jf0.´/j2�d Gf .D/

�f .´/

�:

I GDn t.´/ is a martingale in t so long as t stays away from ´.

(Here t D Œ0; t �.)

1. The pinned measure is two-sided radial SLE

We tilt chordal SLE� by the martingale GDn t.´/, in the sense of the

Girsanov theorem.

The result is a finite measure �´ that is related to chordal SLE� ,considered as a measure on curves stopped before approaching ´, by

d�´

d�. t / D GDn t

.´/:

Under the new measure �´, the curves hit ´ with probability 1.

We complete the curves by tracing a chordal SLE� to the terminalpoint after hitting ´.

The new measure �´ is called two-sided radial SLE� through ´, and itis essentially chordal SLE� conditioned to pass through ´.

2. The length of an SLE curve is its Minkowski contentFor chordal SLE� , the d -dimensional Minkowski content

‚t D Contentd . t / D lim�!0C

�d�2 Areaf´ W dist.´; t / < �g

exists for all t .

The curve has the natural parametrization if ‚t D t for all t .

This parametrization is natural because it transforms like ad -dimensional volume measure under a conformal map f : the timetaken to traverse .f ı /Œ0; t � isZ t

0

ˇf 0� .t/

�ˇddt:

If we parametrize all our SLE� curves by Minkowski content, we mayconstruct the aggregate measure

� D

ZD

�´ dA.´/:

Length-biased chordal SLE

We can prove that the relationship seen in the discrete case is repeatedin the continuum for SLE.

Theorem (F.) Let � � 4. Recall that, in a fixed domain,

I � is chordal SLE� ;

I �´ is two-sided radial SLE� through ´;

I � D

ZD

�´ dA.´/ is the aggregate.

Thend�. / D j j d�. /;

where the length j j D ‚1 is the d -dimensional Minkowski contentof .

In other words, the aggregate of two-sided radial SLE is length-biasedchordal SLE.

An escape estimateOne of the most important estimates needed for this theorem is auniform estimate for the probability that an SLE curve retreats farfrom its target point after first coming very close.

This estimate is new in the case of two-sided radial SLE:

Theorem (F., Lawler). If � � 4, there exists c <1 such that if is atwo-sided radial SLE� curve through 0 in the unit disk,

� D infft > 0 W j .t/j D rg;

�s D infft > � W j .t/j D rsg;

and G is the � -algebra generated by up to time �, then

Pf�s < �0 j Gg � c s�.8=��1/=2:

This says that the probability of escaping, to a distance s times largerthan the closest distance reached, decreases as a power of s.

Methods used

The boundary intersection exponent for SLE� is 8=� � 1: theprobability of passing within � of a smooth boundary point is of order�8=��1.

To prove escape estimates, we use boundary estimates for theprobability of hitting a given crosscut of the domain.

There are typically many crosscuts to consider. To add up theestimates and get a useful bound we need � � 4, so that 8=� � 1 � 1.

To estimate probabilities for two-sided radial SLE, we must estimatethe harmonic measure from ´ of the “far side” of the curve.

The harmonic measure can change dramatically even though thedistance from ´ does not.

Controlling for this issue is a major part of the proof.

Thank you!