On the scaling limit of planar self-avoiding walk. (English) Zbl 1069.60089

Lawler, Gregory F.; Schramm, Oded; Werner, Wendelin

Lapidus, Michel L. (ed.) et al., Fractal geometry and applications: A jubilee of Benoît Mandelbrot. Multifractals, probability and statistical mechanics, applications. In part the proceedings of a special session held during the annual meeting of the American Mathematical Society, San Diego, CA, USA, January 2002. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3638-2/v.2; 0-8218-3292-1/set). Proceedings of Symposia in Pure Mathematics 72, Pt. 2, 339-364 (2004).

The authors deal with a planar self-avoiding random walk (nearest neighbor random walk) in the square lattice with no self-interaction by the use of a new process called the stochastic Loewner evolution which is essentially a one parameter family of conformally invariant processes, indexed by \(n\). Such processes are generated from the one-dimensional Brownian motion, with the parameter \(n\) being a scale. The authors provide precise formulations of the existence of the scaling limit conjectures. They show that if the scaling limit exists and is conformally covariant, then the scaling limit can be indexed by the parameter of the Loewner process. Many of the formulation of self avoiding random walk problems are brought within the ambit of the Loewner processes.
For the entire collection see [Zbl 1055.37003].

Reviewer: S. K. Srinivasan (Chennai)

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
82B41	Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
60G18	Self-similar stochastic processes

Keywords:

nearest neighbour; lattice; stochastic Loewner evolution Brownian motion