Supercritical self-avoiding walks are space-filling

TL;DR

This paper demonstrates that supercritical self-avoiding walks in a lattice become space-filling in the scaling limit, revealing a phase transition in their geometric behavior.

Contribution

It establishes that self-avoiding walks are space-filling in the supercritical regime, providing new insights into their geometric properties in the scaling limit.

Findings

Self-avoiding walks are space-filling when supercritical.

The phase transition occurs at the critical parameter mu.

The behavior differs from subcritical regimes, which are not space-filling.

Abstract

We consider random self-avoiding walks between two points on the boundary of a finite subdomain of Z^d (the probability of a self-avoiding trajectory gamma is proportional to mu^{-length(gamma)}). We show that the random trajectory becomes space-filling in the scaling limit when the parameter mu is supercritical.