The exploration process of critical Boltzmann planar maps decorated by a triangular $O(n)$ loop model

TL;DR

This paper studies the exploration process of critical Boltzmann planar maps decorated with a triangular $O(n)$ loop model, revealing a boundary-conditioned random walk representation and establishing scaling limits and equivalences in the model.

Contribution

It introduces a new boundary condition for the exploration process and derives the scaling limit, connecting it to a self-similar Markov process, advancing understanding of decorated planar maps.

Findings

Law in the non-generic critical phase is coded by a boundary-conditioned random walk.

The exploration process has the same scaling limit as in the bipartite loop model.

Proves the equivalence of admissible weight sequences via the fixed point equation.

Abstract

In this paper we investigate pointed $(\mathbf{q}, g, n)$-Boltzmann loop-decorated maps with loops traversing only inner triangular faces. Using the peeling exploration of arXiv:1809.02012 modified to this setting we show that its law in the non-generic critical phase can be coded in terms of a random walk confined to the positive integers by a new specific boundary condition. Under a technical assumption that we believe to be true, combining this observation with explicit quantities for the peeling law we derive the large deviations property for the distribution of the so-called nesting statistic and show that the exploration process possesses exactly the same scaling limit as in the rigid loop model on bipartite maps that is a specific self-similar Markov process introduced in arXiv:1809.02012. Besides, we conclude the equivalence of the admissible weight sequences related by the so-called fixed point equation by proving the missing direction in the argument of arXiv:1202.5521.