Self-Avoiding Walks on the UIPQ

  • Alessandra Caraceni
  • Nicolas CurienEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 300)


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 the estimates of [15], 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 [9] 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.


Uniform Infinite Planar Quadrangulation Random planar maps Self-avoiding walk Peeling process 



We thank Jérémie Bouttier for fruitful discussion as well as for providing us with an alternative derivation of (1) based on [8]. We are also grateful to Jason Miller for a discussion about [18, 19] and Sect. 5. Figure 1 has been done via Timothy Budd’s software.


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© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of BathBathUK
  2. 2.Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-SaclayOrsayFrance

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