Polygons and the Lace Expansion

Abstract

The lace expansion was introduced by Brydges and Spencer in 1985 [7] to analyse weakly self-avoiding walks in dimensions d > 4. Subsequently it has been generalised and greatly extended, so that it now applies to a variety of problems of interest in probability theory, statistical physics, and combinatorics, including the strictly self-avoiding walk, lattice trees, lattice animals, percolation, oriented percolation, the contact process, random graphs, and the Ising model. A recent survey is [42]. In this chapter, we give an introduction to the lace expansion for self-avoiding walks, with emphasis on self-avoiding polygons. We focus on combinatorial rather than analytical aspects. © 2009 Cano..