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.

Type

Publication

Date

September, 2018

Links

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