Trees of self-avoiding walks

TL;DR

This paper studies biased random walks on trees derived from self-avoiding walks, proposing a new way to construct measures on infinite self-avoiding walks and analyzing their properties near critical bias.

Contribution

It introduces a novel approach to define measures on infinite self-avoiding walks via biased random walks on trees and provides criteria for the continuity of escape probabilities.

Findings

Established a criterion for the continuity of escape probabilities.

Proved stability of escape probability functions under uniform convergence.

Conjectured the limit measure coincides with the weak limit of uniform SAW.

Abstract

We consider the biased random walk on a tree constructed from the set of finite self-avoiding walks on a lattice, and use it to construct probability measures on infinite self-avoiding walks. The limit measure (if it exists) obtained when the bias converges to its critical value is conjectured to coincide with the weak limit of the uniform SAW. Along the way, we obtain a criterion for the continuity of the escape probability of a biased random walk on a tree as a function of the bias, and show that the collection of escape probability functions for spherically symmetric trees of bounded degree is stable under uniform convergence.