Weighted self-avoiding walks

TL;DR

This paper investigates the properties of weighted self-avoiding walks on infinite graphs, establishing conditions for the equality of connective and bridge constants and proving a continuity theorem for these constants relative to weight functions.

Contribution

It introduces conditions under which weighted connective and bridge constants are equal and proves a continuity theorem for connective constants with respect to weight functions.

Findings

Equality of weighted connective and bridge constants under certain conditions

Continuity of connective constants with respect to changes in weight functions

Handling unbounded support of weight functions through generalized walk length

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.