Self-avoiding walk is sub-ballistic

TL;DR

This paper proves that self-avoiding walks on integer lattices in two or more dimensions do not exhibit ballistic behavior, meaning they grow slower than linearly with high probability.

Contribution

It establishes the sub-ballistic nature of self-avoiding walks in all dimensions d ≥ 2, a significant theoretical result in understanding their large-scale behavior.

Findings

Self-avoiding walks are sub-ballistic in all dimensions d ≥ 2.

Probability of walks reaching linear distance decays exponentially.

Provides rigorous proof of sub-ballistic growth rate.

Abstract

We prove that self-avoiding walk on Z^d is sub-ballistic in any dimension d at least two. That is, writing ||u|| for the Euclidean norm of u \in Z^d, and SAW_n for the uniform measure on self-avoiding walks gamma:{0,...,n} \to Z^d for which gamma_0 = 0, we show that, for each v > 0, there exists c > 0 such that, for each positive integer n, SAW_n (max {|| gamma_k || : k \in {0,...,n}} > v n) < e^{- c n}.