Large deviation Local Limit Theorems and limits of biconditioned Trees and Maps - Archive ouverte HAL
Article Dans Une Revue The Annals of Applied Probability Année : 2023

Large deviation Local Limit Theorems and limits of biconditioned Trees and Maps

Résumé

We first establish new local limit estimates for the probability that a nondecreasing integer-valued random walk lies at time $n$ at an arbitrary value, encompassing in particular large deviation regimes. This enables us to derive scaling limits of such random walks conditioned by their terminal value at time $n$ in various regimes. We believe both to be of independent interest. We then apply these results to obtain invariance principles for the Lukasiewicz path of Bienaym\'e-Galton-Watson trees conditioned on having a fixed number of leaves and of vertices at the same time, which constitutes a first step towards understanding their large scale geometry. We finally deduce from this scaling limit theorems for random bipartite planar maps under a new conditioning by fixing their number of vertices, edges, and faces at the same time. In the particular case of the uniform distribution, our results confirm a prediction of Fusy & Guitter on the growth of the typical distances and show furthermore that in all regimes, the scaling limit is the celebrated Brownian map.

Dates et versions

hal-03371111 , version 1 (08-10-2021)

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Citer

Igor Kortchemski, Cyril Marzouk. Large deviation Local Limit Theorems and limits of biconditioned Trees and Maps. The Annals of Applied Probability, 2023, 33 (5), ⟨10.1214/22-AAP1906⟩. ⟨hal-03371111⟩
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