SLE revision 2: Loewner’s Differential Equation

Last time I set up the geometric notions of probability that will be needed to proceed with the course material. Now we consider the deterministic differential equation due to Loewner (1923) which he used to make progress on the Bieberbach Conjecture, but which will also underpin the construction of SLE. This proof is adapted for this specific case from the slightly more general argument in Duren’s Univalent Functions (Section 3.3). Because in that setting the result concerns an infinite domain, readers should beware that though I am using identical notation, in about half the cases, my function are the inverse and my sets the complement of what they are in Duren.

To explain the construction, as with so many things, a picture speaks a thousand words. Unfortunately I have neither the software nor, right now, the time to produce the necessary diagrams, so the following will have to suffice.

Consider a deterministic simple curve in the unit disc, $(\gamma(t): t\in[0,\infty))$. Removing initial segments of the curve gives the nested simply-connected regions:

$U_t:=\mathbb{U}\backslash \gamma[0,t]$.

Then define as in the previous post, the unique conformal map

$f_t: U_s\rightarrow \mathbb{U}$ such that $f_t(0)=0, f_t'(0)\in\mathbb{R}^+$,

and furthermore set $\xi_t$ to be the image of $\gamma(t)$ under this map. (Note that though the conformal map is not defined on the boundary, it must extend continuously).

$f_t'(0)$ is increasing.

Very informally, this derivative records how much twisting is required at the origin to turn the slit domain into the open disk. Extending the path will demand further twisting. More rigorously, set:

$g_t=f_t^{-1}:\mathbb{U}\rightarrow U_t$.

Then the $(g_t)$ are injective functions from the unit disc to itself which preserve the origin, so Schwarz’s lemma applies. They are clearly not rotations, so

$|g_t'(0)|<1$.

By the inverse function theorem, $|f_t'(0)|>1$ (*). Now, given $t>s$, can decompose:

$f_t=f_s\circ \tilde{f}$,

and $\tilde{f}$ has this useful Schwarz property (*) also. By applying the chain rule, noting that $f_s(0)=0$, we deduce that

$|f_t'(0)|>|f_s'(0)|$.

This means we are free to demand that the curve has time parameter such that

$|f_t'(0)|=e^t$.

A reminder of the statement of Schwarz’s Lemma: Continue reading

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.

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. Continue reading