LARGE DEVIATION LOCAL LIMIT THEOREMS AND LIMITS OF BICONDITIONED PLANAR MAPS

• 成果类型: Article
• 署名作者: Kortchemski, Igor; Marzouk, Cyril
• 署名单位: Institut Polytechnique de Paris; Ecole Polytechnique; Centre National de la Recherche Scientifique (CNRS); Institut Polytechnique de Paris; Ecole Polytechnique
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/22-AAP1906
• 发表日期: 2023
• 页码: 3755-3802
• 关键词: galton-watson trees; independent random vectors; SCALING LIMITS; INVARIANCE-PRINCIPLES; superlarge deviations; cramers condition; random-walks; sums; number; asymptotics

摘要

We first establish new local limit estimates for the probability that a non -decreasing integer-valued random walk lies at time n at an arbitrary value, encompassing in particular large deviation regimes on the boundary of the Cramer zone. 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 in-variance principles for the Lukasiewicz path of Bienayme-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 and Guitter on the growth of the typical distances and show furthermore that in all regimes, the scaling limit is the celebrated Brownian sphere.