Abstract
We prove some asymptotic results for the radius and the profile of large random planar maps with faces of arbitrary degrees. Using a bijection due to Bouttier, Di Francesco & Guitter between rooted planar maps and certain four-type trees with positive labels, we derive our results from a conditional limit theorem for four-type spatial Galton-Watson trees.
Highlights
This paper is devoted to the proof of limit theorems for random planar maps with no constraint on the degree of faces
This work is a natural sequel to the papers [2, 4, 8, 12, 11], which dealt with such limit theorems with an increasing level of generality, starting from the case of planar quadrangulations and moving to invariance principles for the radius and the profile of bipartite, general, Boltzmann-distributed random planar maps
We obtain thanks to Lemma 3.9 and Proposition 3.12 that for every s ∈ [0, 1], there exists a constant ε′ > 0 such that for all n sufficiently large, P(μ1,)←→ ,νn,1 |Gt1(⌊ns⌋) − a−1 1ns| ≥ n3/4 ≤ e−nε
Summary
This paper is devoted to the proof of limit theorems for random planar maps with no constraint on the degree of faces. This work is a natural sequel to the papers [2, 4, 8, 12, 11], which dealt with such limit theorems with an increasing level of generality, starting from the case of planar quadrangulations and moving to invariance principles for the radius and the profile of bipartite, general, Boltzmann-distributed random planar maps. Our main goal is to obtain invariance principles for certain functionals of planar maps with no constraint on the face degrees, of the same kind as those obtained in [11]. Our approach in this paper will be to focus essentially on these differences, while the parts which can be derived mutatis mutandis from [4, 12] will be omitted
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