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Self-Avoiding Walks on the UIPQ

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Sojourns in Probability Theory and Statistical Physics - III

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 300))

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 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.

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Notes

  1. 1.

    Remark that this notation is not coherent with that of [9], where we denoted by \(\mathcal {H}_\infty \) an object with a boundary that is not necessarily simple, and by \(\widetilde{\mathcal {H}}_\infty \) the one that is central to this paper, obtained from \(\mathcal {H}_\infty \) by a pruning procedure. Since the general boundary UIHPQ will make no appearance in this paper, we shall drop the tilde with no fear of confusion.

  2. 2.

    Actually the works [9, 24, 25] show a convergence in distribution and one needs to prove uniform integrability to be able to pass to the expectation. We do not give the details since the actual proof bypasses this technical issue.

  3. 3.

    Notice that this is in contrast with the notation of [9], where a distinction needed to be made between quadrangulations with a general boundary and ones whose boundary was required to be simple, which we usually signalled with a “tilde” over the relevant symbol.

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Acknowledgments

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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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Bath, Bath, UK

    Alessandra Caraceni

  2. Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405, Orsay, France

    Nicolas Curien

Authors
  1. Alessandra Caraceni
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  2. Nicolas Curien
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Corresponding author

Correspondence to Nicolas Curien .

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Editors and Affiliations

  1. NYU Shanghai, Shanghai, China

    Vladas Sidoravicius

Additional information

We dedicate this work to Chuck Newman on the occasion of his 70th birthday

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Caraceni, A., Curien, N. (2019). Self-Avoiding Walks on the UIPQ. In: Sidoravicius, V. (eds) Sojourns in Probability Theory and Statistical Physics - III. Springer Proceedings in Mathematics & Statistics, vol 300. Springer, Singapore. https://doi.org/10.1007/978-981-15-0302-3_5

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