An introduction to the stochastic Loewner evolution. (English) Zbl 1061.60107

Lawler, Gregory F.

Kaimanovich, Vadim A. (ed.), Random walks and geometry. Proceedings of a workshop at the Erwin Schrödinger Institute, Vienna, June 18 – July 13, 2001. In collaboration with Klaus Schmidt and Wolfgang Woess. Collected papers. Berlin: de Gruyter (ISBN 3-11-017237-2/hbk). 261-293 (2004).

The stochastic Loewner evolution is a one-parameter family of conformally invariant measures on curves in the plane. As a family of growing clusters it is conjectured to be the scaling limit of many statistical physics lattice models, like e.g. random walks, self-avoiding random walks, percolation, loop erased walks, Potts models, interface growth phenomena. The present paper is of an expository nature and is intended to give an introduction to the stochastic Loewner evolution for probabilists. After introducing three basic deterministic Loewner equations (chordal, radial and whole plane), their stochastic version is investigated by exploiting the driving by the Brownian motion with a suitable variance parameter. A list of hints is given towards discrete, physics motivated models, whose scaling limits share conformal invariance property. The method for calculating scaling exponents is elaborated and next applied to find exponents for Brownian excursions. Statistical physics argument relating partition functions to martingales can be used to get a condition under which random geometric processes can describe interfaces in two-dimensional statistical mechanics at criticality. Applications to percolation and the Ising model can be found in the archived preprint by M. Bauer, D. Bernard and Kytölä [“Multiple Schramm-Loewner evolutions and statistical mechanics martingales” (arXiv:math-ph/0503024)].
For the entire collection see [Zbl 1047.60001].

Reviewer: Piotr Garbaczewski (Zielona Góra)

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
82B41	Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
30C35	General theory of conformal mappings
58J65	Diffusion processes and stochastic analysis on manifolds

Keywords:

Loewner equation; conformal field theory; scaling limits; critical exponents; Brownian excursions