Self-Avoiding Walks on the 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 (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.
Keywords
Cite
@article{arxiv.1609.00245,
title = {Self-Avoiding Walks on the UIPQ},
author = {Alessandra Caraceni and Nicolas Curien},
journal= {arXiv preprint arXiv:1609.00245},
year = {2017}
}