# Extendable self-avoiding walks

### Geoffrey R. Grimmett

University of Cambridge, UK### Alexander E. Holroyd

Microsoft Research, Redmond, USA### Yuval Peres

Microsoft Research, Redmond, USA

## 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.

## Cite this article

Geoffrey R. Grimmett, Alexander E. Holroyd, Yuval Peres, Extendable self-avoiding walks. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 1 (2014), no. 1, pp. 61–75

DOI 10.4171/AIHPD/3