SLE revision 1: Properties of Random Sets

Prof. Werner’s excellent Part III course ‘Topics in Conformal Invariance and Randomness’ has recently finished, and I’ve been doing some revision. The course begins with a general discussion of some of the ideas useful in demanding some form of regularity for random paths or random sets in a domain. For example, for continuous time processes, we can define a Markovian property: this is both easy and natural, mainly because the state space, assuming it is $\mathbb{R}^d$ is homogenous, which is not a luxury in, say, the unit disc. In two dimensions, things are particularly tractable because of the equivalence to the complex plane, and from this we develop the Schramm-Loewner evolution, and we examine its properties. In particular, SLEs with some exponents arise as a limit of discrete processes, with wide-ranging applications. In this first note, we motivate and explain some properties that we might wish random sets to have.

A conformal map is an invertible map between domains in the complex plane which preserves angles. Riemann’s mapping theorem states that there exists a conformal map from any non-empty, simply connected domain to the open unit disc. We have some freedom to control one point, and the boundary is mapped to the boundary.

Conformal Invariance: Given a simply connected domain D, and conformal $\phi:D\rightarrow\mathbb{U}$ , then $\mathcal{B}^D$ a process defined on domains D is conformally invariant if

$\phi(\mathcal{B}^D)\stackrel{d}{=}\mathcal{B}^\mathbb{U}$ .

This says that the law of the process is preserved under the transformation.

The notation chosen is deliberate. The best example is Brownian paths: take B a Brownian motion started at 0, and $T^D$ the exit time of domain D, then set $\mathcal{B}=\{B_t,t\leq T^D\}$ the path in D. Informally, conformal invariance for all domains with $0\in D$ , follows because BM is isotropic, that is, the angle taken after a time t, whatever that means, is uniformly distributed. Modulo Markov technicalities, this property is preserved under a conformal map because they preserve angles.

Conformal Restriction: This is essentially the same as conformal invariance, but in the special case where one of the domains is contained in the other. Although less general, by viewing everything in the context of the laws of processes in the larger domain, we can in fact show an equality for a given single process with conditioning, rather than effectively two unrelated processes. We assume the reference domain is the unit disc.

Concretely, we can consider a random set K in the unit disc with law $P^K$ , and for a subset $U\subset\mathbb{U}$ which contains 0 and 1, define the conformal map $\phi_U:U\rightarrow \mathbb{U}$ that preserves 0 and 1. Then set $P_U^K$ to be the law of $\phi_U^{-1}(K)$ , which gives a law for random sets in U. We say K satisifies conformal restriction if:

$P_U^K=P|_{\{K\subset U\}}$

Observe that applying $\phi_U$ to both sides of the definition gives conformal invariance for this pair of domains.

This is useful, because the set of such conformal maps form a semi-group. We will often want to describe the law of a random set by the probabilities of it lying within subsets of the domain: $A(\phi_U):= P^K(K\subset U)$ , and conformal restriction implies that

$A(\phi_{U_1}\circ\phi_{U_2})=A(\phi_{U_1})A(\phi_{U_2})$ .

Now it can be shown that for Brownian Motion in the unit disc, multiplied by a random variable to ensure it leaves the disc at 1:

$Y=\left(\frac{B_t}{B_{T_1}},t\leq T_1\right)$ ,

the relevant law is specified by

$\mathbb{P}(Y\subset U)=\phi_U'(1)$ .

This motivates the idea that

$A(\phi_U)=|\phi_U'(0)|^\alpha \phi_U'(1)^\beta$

would be an excellent form for A. The question is: for which values of $\alpha,\beta$ can we define a random set with this property? Some immediate restrictions will apply to ensure that A is in fact a probability.

Conformal Markov Property: Now consider a random curve $\gamma$ . We want this to have as much conformal restriction as is useful, but perhaps demanding conformal restriction on the entire domain is too strong? A trite analogue might be that a Markov chain might have more than one component, but we don’t care about its behaviour on a component which because of initial conditions we won’t explore. In particular, the notion of space-homogeneity could be relaxed to ‘space-homogeneity over the subset of the state space that we actually visit’. I’m not sure that makes a whole lot of sense, but there’ll be an example to explain it soon.

We have a definition as for conformal restriction, but it only applies when U is an initial segment of the curve. Formally, $\gamma$ has the conformal Markov property if $\gamma(0)=1,\gamma(\infty)=0$ and:

Conditional on $\gamma[0,t]$ , $\gamma[t,\infty)\stackrel{d}{=}f_t^{-1}(\tilde{\gamma}\xi_t)$

where $f_t$ is the conformal map $\mathbb{U}\backslash\gamma[0,t]\rightarrow\mathbb{U}$ preserving 0; $\tilde{\gamma}$ is an independent copy of $\gamma$ , and $\xi_t=f_t(\gamma_t)$ the image of $\gamma(t)$ lying on the boundary of the unit disc. The last term looks confusing, but really just resets for the fact that the curve has to start from 1, rather than generally on the boundary.

Example: to illustrate the difference between this conformal Markov property and conformal restriction, consider a non-random straight line between 1 and 0. Taking U to be anything other than the disc without a slit along the positive real axis, this clearly does not satisfy conformal restriction. However, thinking about what the $f_t$ s look like, it does satisfy CMP.

Remarks: – The key to this property is the introduction of $\xi$ , a process which is complicated to define, but is easy to work with since it lives on the boundary of the disc, which is a nice region. The next step will be an attempt to reverse the construction – that is, can we build $\gamma$ if we are given $\xi$ ? For example, can we build a framework where conformal Markov can instead be read as:

$\left(\frac{\xi_{T+t}}{\xi_T},t\geq 0\right)\stackrel{d}{=}\tilde{\xi}$ ?

– The ‘glue’ between these two stochastic objects are the conformal transformation functions $f_t$ . Since we are interested in $\gamma$ as a curve, rather than a function or process, we have freedom to reparameterise the paths. In particular, it will turn out to be the case that parameterising such that $f_t'(0)=e^t$ will give a differential equation, the Loewner equation, linking $f_t$ and $\xi_t$ in a highly applicable way.

References: To be added very soon, but mainly just Prof. Werner’s lectures.

Eventually Almost Everywhere

A blog about probability and olympiads by Dominic Yeo

SLE revision 1: Properties of Random Sets

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