Random Walks And Random Self-Avoiding Walks

Abstract

It is well-known that the continuum limit of a random walk on a lattice is Brownian motion. Similarly it is believed (but not known) that the continuum limit of a self-avoiding walk is so called Schramm-Loewner evolution, and that the continuum limit of a random self-avoiding polygon is a random loop measure recently constructed by Wendelin Werner. In this paper we give an exposition of these objects. The main questions we eventually focus on are combinatorial issues regarding numbers of self-avoiding walks and polygons. It is a surprise to find that the connective constant for self-avoiding walks and self-avoiding polygons are the same. Some of these issues have been recently settled by Mardras, Slade, Lawler, Duminil-Copin, Smirnov, and Hammond.