Extendable self-avoiding walks

TL;DR

This paper proves that for certain directed graphs, the growth rates of various types of extendable self-avoiding walks are equal, extending understanding of their connective constants.

Contribution

It establishes the equality of connective constants for forward, backward, and doubly extendable self-avoiding walks on a broad class of directed graphs.

Findings

Connective constants for different extendability types are equal.

Existence of these constants is proven for a wide class of graphs.

Results depend on graph properties like unimodularity.

Abstract

The connective constant mu of a graph is the exponential growth rate of the number of n-step self-avoiding walks starting at a given vertex. A self-avoiding walk is said to be forward (respectively, backward) extendable if it may be extended forwards (respectively, backwards) to a singly infinite self-avoiding walk. It is called doubly extendable if it may be extended in both directions simultaneously to a doubly infinite self-avoiding walk. We prove that the connective constants for forward, backward, and doubly extendable self-avoiding walks, denoted respectively by mu^F, mu^B, mu^FB, exist and satisfy mu = mu^F = mu^B = mu^FB for every infinite, locally finite, strongly connected, quasi-transitive directed graph. The proofs rely on a 1967 result of Furstenberg on dimension, and involve two different arguments depending on whether or not the graph is unimodular.