Weighted self-avoiding walks

Geoffrey R Grimmett,Zhongyang Li

doi:10.1007/s10801-019-00895-6

Abstract

We study the connective constants of weighted self-avoiding walks (SAWs) on infinite graphs and groups. The main focus is upon weighted SAWs on finitely generated, virtually indicable groups. Such groups possess so-called height functions, and this permits the study of SAWs with the special property of being bridges. The group structure is relevant in the interaction between the height function and the weight function. The main difficulties arise when the support of the weight function is unbounded, since the corresponding graph is no longer locally finite. There are two principal results, of which the first is a condition under which the weighted connective constant and the weighted bridge constant are equal. When the weight function has unbounded support, we work with a generalized notion of the ‘length’ of a walk, which is subject to a certain condition. In the second main result, the above equality is used to prove a continuity theorem for connective constants on the space of weight functions endowed with a suitable distance function.

Highlights

The counting of self-avoiding walks (SAWs) is extended here to the study of weighted SAWs
Each edge is assigned a weight, and the weight of a SAW is defined as the product of the weights of its edges
We study certain properties of the exponential growth rate μ in terms of the weight function φ, including its continuity on the space of weight functions

Summary

Introduction

The counting of self-avoiding walks (SAWs) is extended here to the study of weighted SAWs. One of the main technical steps in the current work is the proof, subject to certain conditions, of the equality of the weighted connective constant and the weighted bridge constant This was proved in [12, Thm 4.3] for the unweighted constants on any connected, infinite, quasi-transitive, locally finite, simple graph possessing a unimodular graph height function.. This was proved in [12, Thm 4.3] for the unweighted constants on any connected, infinite, quasi-transitive, locally finite, simple graph possessing a unimodular graph height function.1 This result is extended here to weighted SAWs on finitely generated, virtually indicable groups We write R for the reals, Z for the integers, and N for the natural numbers

Weighted walks and bridges on locally finite graphs

Weight functions with unbounded support

Continuity of the connective constant