The lace expansion on a tree with application to networks of self-avoiding walks

Abstract

The lace expansion has been used successfully to study the critical behaviour in high dimensions of self-avoiding walks, lattice trees and lattice animals, and percolation. In each case, the lace expansion has been an expansion along a time interval. In this paper, we introduce the lace expansion on a tree, in which ‘time’ is generalised from an interval to a tree. We develop the expansion in the context of networks of mutually-avoiding self-avoiding walks joined together with the topology of a tree, in dimensions d>4, and prove Gaussian behaviour for sufficiently spread-out networks consisting of long self-avoiding walks.