Self-Avoiding Walks on the UIPQ

TL;DR

This paper investigates a model of self-avoiding walks on the uniform infinite planar quadrangulation, demonstrating diffusive behavior and revealing that the model's law is singular compared to the standard UIPQ, indicating persistent disorder.

Contribution

It establishes a lower bound on the displacement of the self-avoiding walk and shows the walk is diffusive, also proving the model's law is singular relative to the standard UIPQ.

Findings

Self-avoiding walk is diffusive on UIPQ.

Volume growth exponent is 4.

Model's law is singular with respect to standard UIPQ.

Abstract

We study an annealed model of Uniform Infinite Planar Quadrangulation (UIPQ) with an infinite two-sided self-avoiding walk (SAW), which can also be described as the result of glueing together two independent uniform infinite quadrangulations of the half-plane (UIHPQs). We prove a lower bound on the displacement of the SAW which, combined with estimates from our previous paper, shows that the self-avoiding walk is diffusive. As a byproduct this implies that the volume growth exponent of the lattice in question is $4$ (as is the case for the standard UIPQ); nevertheless, using our previous work we show its law to be singular with respect to that of the standard UIPQ, that is -- in the language of statistical physics -- the fact that disorder holds.