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08 February 2025
 
  » arxiv » 1609.0245

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Self-Avoiding Walks on the UIPQ
Alessandra Caraceni ; Nicolas Curien ;
Date 1 Sep 2016
AbstractWe 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.
Source arXiv, 1609.0245
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