Self-avoiding walk on nonunimodular transitive graphs

TL;DR

This paper proves that self-avoiding walks on certain nonunimodular transitive graphs exhibit ballistic behavior, exponential decay of the two-point function at criticality, and mean-field critical exponents, extending known results to a broader class of graphs.

Contribution

It establishes new results on self-avoiding walks on nonunimodular transitive graphs, including ballisticity and decay properties, and extends these findings to related repulsive walk models.

Findings

Self-avoiding walk is ballistic on these graphs.

Critical two-point function decays exponentially.

Critical exponent for susceptibility is mean-field.

Abstract

We study self-avoiding walk on graphs whose automorphism group has a transitive nonunimodular subgroup. We prove that self-avoiding walk is ballistic, that the bubble diagram converges at criticality, and that the critical two-point function decays exponentially in the distance from the origin. This implies that the critical exponent governing the susceptibility takes its mean-field value, and hence that the number of self-avoiding walks of length $n$ is comparable to the $n$th power of the connective constant. We also prove that the same results hold for a large class of repulsive walk models with a self-intersection based interaction, including the weakly self-avoiding walk. All these results apply in particular to the product $T_k \times \mathbb{Z}^d$ of a $k$-regular tree ($k\geq 3$) with $\mathbb{Z}^d$, for which these results were previously only known for large $k$.