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Extendable self-avoiding walks

Accepted version
Peer-reviewed

Type

Article

Change log

Authors

Grimmett, Geoffrey R 
Holroyd, Alexander E 
Peres, Yuval 

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.

Description

Keywords

math.CO, math.CO, math-ph, math.MP, 05C30, 82B20, 60K35

Journal Title

Annales de l’Institut Henri Poincaré D

Conference Name

Journal ISSN

2308-5827
2308-5835

Volume Title

1

Publisher

European Mathematical Society

Rights

All rights reserved
Sponsorship
Engineering and Physical Sciences Research Council (EP/I03372X/1)