Positive speed self-avoiding walks on graphs with more than one end

TL;DR

This paper proves that on certain infinite graphs with multiple ends, self-avoiding walks exhibit positive speed and linear end-to-end distance growth, with the mean square displacement exponent being 1, advancing understanding of SAWs on complex structures.

Contribution

It establishes that on graphs with more than one end, SAWs have positive speed and linear growth, using novel applications of Kesten's pattern theorem and Stallings' splitting theorem.

Findings

SAWs have positive speed on graphs with more than one end.

The mean square displacement exponent is 1 on such graphs.

SAWs exhibit exponential growth rate comparable to all walks.

Abstract

A self-avoiding walk (SAW) is a path on a graph that visits each vertex at most once. The mean square displacement of an $n$-step SAW is the expected value of the square of the distance between the ending point and the starting point of an $n$-step SAW, where the expectation is taken with respect to the uniform measure on $n$-step SAWs starting from a fixed vertex. It is conjectured that the mean square displacement of an $n$-step SAW is asymptotically $n^{2\nu}$, where $\nu$ is a constant. Computing the exact values of the exponent $\nu$ on various graphs has been a challenging problem in mathematical and scientific research for long. In this paper we show that on any locally finite Cayley graph of an infinite, finitely-generated group with more than two ends, the number of SAWs whose end-to-end distances are linear in lengths has the same exponential growth rate as the number of all the SAWs. We also prove that for any infinite, finitely-generated group with more than one end, there exists a locally finite Cayley graph on which SAWs have positive speed - this implies that the mean square displacement exponent $\nu=1$ on such graphs. These results are obtained by proving more general theorems for SAWs on quasi-transitive graphs with more than one end, which make use of a variation of Kesten's pattern theorem in a surprising way, as well as the Stalling's splitting theorem. Applications include proving that SAWs have positive speed on the square grid in an infinite cylinder, and on the infinite free product graph of two connected, quasi-transitive graphs.