On the probability that self-avoiding walk ends at a given point

TL;DR

This paper proves that the probability of a uniform self-avoiding walk ending at a specific point diminishes as the walk length increases, indicating delocalization of the endpoint in high dimensions.

Contribution

It establishes uniform decay bounds for the endpoint probability of self-avoiding walks on Z^d, advancing understanding of their endpoint distribution behavior.

Findings

Probability tends to 0 as walk length increases

Decay rate faster than n^{-1/4 + epsilon} for fixed points

Bounds on the probability of the walk forming a polygon when |x|=1

Abstract

We prove two results on the delocalization of the endpoint of a uniform self-avoiding walk on Z^d for d>1. We show that the probability that a walk of length n ends at a point x tends to 0 as n tends to infinity, uniformly in x. Also, for any fixed x in Z^d, this probability decreases faster than n^{-1/4 + epsilon} for any epsilon >0. When |x|= 1, we thus obtain a bound on the probability that self-avoiding walk is a polygon.