SLE and conformal invariance (following Lawler, Schramm and Werner). (SLE et invariance conforme (d’après Lawler, Schramm et Werner).) (French) Zbl 1083.60067

Bertoin, Jean

Bourbaki seminar. Volume 2003/2004. Exposes 924–937. Paris: Société Mathématique de France (ISBN 2-85629-173-2/pbk). Astérisque 299, 15-28, Exp. No. 925 (2005).

The stochastic Loewner evolution (SLE) has improved the understanding of the critical behavior of two-dimensional systems from statistical physics, in particular, it presents a rigorous approach to understand the asymptotic behaviour of discrete models involving conformal invariant and self-avoiding or loop-erased discrete random walks. The aim of this paper is to give a nontechnical introduction to this subject. The deterministic Loewner equation and its stochastic extension, the SLE are introduced. Then some crucial properties for particular values of the parameter are anounced. Finally, the last part presents several situations, where SLE permits concretely to construct some remarkable curves in the plane. For a more complete presentation, the author refers to W. Werner [in: Lectures on probability theory and statistics. Lect. Notes Math. 1840, 109–195 (2004; Zbl 1057.60078)] and G. F. Lawler [in: Random walks and geometry, 261–293 (2004; Zbl 1061.60107)].
For the entire collection see [Zbl 1066.00008].

Reviewer: Cathérine Rainer (Brest)

MSC:

60J65	Brownian motion
60K35	Interacting random processes; statistical mechanics type models; percolation theory
82B43	Percolation
30C35	General theory of conformal mappings

Keywords:

Stochastic Loewner equation; planar Brownian motion; percolation; random walk; scaling limit

Citations:

Zbl 1057.60078; Zbl 1061.60107