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

TL;DR

This paper develops new local limit estimates for integer-valued random walks, enabling the derivation of scaling limits for conditioned trees and maps, ultimately showing their convergence to the Brownian map in various regimes.

Contribution

It introduces novel local limit theorems for random walks and applies them to establish scaling limits of conditioned trees and maps, advancing understanding of their large-scale geometry.

Findings

New local limit estimates for nondecreasing integer-valued random walks.

Scaling limits of conditioned Bienaymé-Galton-Watson trees are derived.

Random bipartite planar maps converge to the Brownian map in various regimes.

Abstract

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.