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

• Published in: The Annals of applied probability Vol. 33; no. 5
• Main Authors: Kortchemski, Igor, Marzouk, Cyril
• Format: Journal Article
• Language: English
• Published: Institute of Mathematical Statistics (IMS) 01-10-2023
• Subjects: Mathematics; Probability
• ISSN: 1050-5164; 2168-8737
• DOI: 10.1214/22-AAP1906

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.