The Schramm-Löwner Evolation
Specialeforsvar ved Mads Bonde Raad
Titel: The Schramm-Löwner Evolation
Resume: Colloquially speaking, The SLE may be described as a distribution on a system of information. The system is a soft word for four abstract mathematical elements, each able to generate the remaining three. In this paper we shall discuss each of the four ways of describing the SLE, and describe the relation to the other elements.
First we establish theory concerning families of compact hulls in the complex halfplane
H. A compact hull Kt is a bounded set in H, whose complement in H is open and simply connected. The SLE on compact hulls is a distribution on a nicely-behaved subset of hull-families, having the so-called Local Growth Property. Any increasing family of compact hulls induces two new elements. The first is a driving function, which is a continuous function : [0;1) ! R. The SLE as a distribution of 's is simply a Brownian motion with a variance parameter traditionally denoted as . The second induced element is a Loewner ow (gt) made of what we shall call mapping-out functions. These are conformal maps from
Ht := HnKt to H, made unique by ensuring that gt z for large moduli. It turns out that allows inversion; it generates the corresponding Loewner ow and in turn the LGP family. The rst is retrived as the unique solution to the Loewner differential equation given by
@tgt (z) =2gt (z) (t); g0 (z) = z:
The second is realized to be Kt = fz 2 H : t (z)g where (z) denotes the maximal lifetime of the solution to the Loewner equation and bears the name swallowing time of z. Even though there exist examples in the literature of inverse flows g1t that do not extend continuously to
z2 R, it is almost surely the case under the SLE law. The SLE trace may then be de ned as t= g1t( (t)). We may retrieve Kt as all bounded path-components of Hn [0; t] ; and therefore the trace carries full information of the system as well. We shall also discuss the phases of the SLE, which is three sets of inducing different properties for the trace. Finally, we pursue some visualization through simulation of the process.
Vejledere: Bergfinnur Durhuus, Jan Philip Solovej
Censor: Jacob Schach Møller, Aarhus Universitet