The peeling process on random planar maps coupled to an O(n) loop model (with an appendix by Linxiao Chen)

Abstract

We extend the peeling exploration introduced in [Budd (2015)] to the setting of Boltzmann planar maps coupled to a rigid O(n) loop model. Its law is related to a class of discrete Markov processes obtained by confining random walks to the positive integers with a new type of boundary condition. As an application we give a rigorous justification of the phase diagram of the model presented in [Borot, Bouttier, Guitter (2011)]. This entails two results pertaining to the so-called fixed-point equation: the first asserts that any solution determines a well-defined model, while the second result, contributed by Chen in the appendix, establishes precise existence criteria.

A scaling limit for the exploration process is identified in terms of a new class of positive self-similar Markov processes, going under the name of ricocheted stable processes. As an application we study distances on loop-decorated maps arising from a particular first passage percolation process on the maps. In the scaling limit these distances between the boundary and a marked point are related to exponential integrals of certain Lévy processes. The distributions of the latter can be identified in a fairly explicit form using machinery of positive self-similar Markov processes.

Finally we observe a relation between the number of loops that surround a marked vertex in a Boltzmann loop-decorated map and the winding angle of a simple random walk on the square lattice. As a corollary we give a combinatorial proof of the fact that the total winding angle around the origin of a simple random walk started at $(p,p)$ and killed upon hitting $(0,0)$ normalized by $\log p$ converges in distribution to a Cauchy random variable.

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Citations

  1. Borot, Gaëtan, Jérémie Bouttier, and Bertrand Duplantier. “Nesting statistics in the O(n) loop model on random planar maps.” arXiv preprint arXiv:1605.02239 (2016).
  2. Budd, Timothy, and Nicolas Curien. “Geometry of infinite planar maps with high degrees.” Electronic Journal of Probability 22 (2017).
  3. Chen, Linxiao, Nicolas Curien, and Pascal Maillard. “The perimeter cascade in critical Boltzmann quadrangulations decorated by an $ O (n) $ loop model.” arXiv preprint arXiv:1702.06916 (2017).
  4. Richier, Loïc. “Limits of the boundary of random planar maps.” Probability Theory and Related Fields (2017): 1-39.
  5. Chen, Linxiao. Random Planar Maps coupled to Spin Systems. Diss. Université Paris-Saclay, 2018.
  6. Budd, Timothy, Nicolas Curien, and Cyril Marzouk. “Infinite random planar maps related to Cauchy processes.” arXiv preprint arXiv:1704.05297 (2017).
  7. Albenque, Marie, Laurent Ménard, and Gilles Schaeffer. “Local convergence of large random triangulations coupled with an Ising model.” arXiv preprint arXiv:1812.03140 (2018).