RANDOM PARTITIONS UNDER THE PLANCHEREL-HURWITZ MEASURE, HIGH-GENUS HURWITZ NUMBERS AND MAPS

• 成果类型: Article
• 署名作者: Chapuy, Guillaume; Louf, Baptiste; Walsh, Harriet
• 署名单位: Universite Paris Cite; Centre National de la Recherche Scientifique (CNRS); Uppsala University; Centre National de la Recherche Scientifique (CNRS); Universite Paris Cite; Ecole Normale Superieure de Lyon (ENS de LYON)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/23-AOP1651
• 发表日期: 2024
• 页码: 1225-1252
• 关键词: longest increasing subsequence; Random Permutation; asymptotics; enumeration; EQUATIONS

摘要

We study the asymptotic behaviour of random integer partitions under a new probability law that we introduce, the Plancherel-Hurwitz measure. This distribution, which has a natural definition in terms of Young tableaux, is a deformation of the classical Plancherel measure, which appears naturally in the context of Hurwitz numbers, enumerating certain transposition factorisations in symmetric groups. We study a regime in which the number of factors in the underlying factorisations grows linearly with the order of the group, and the corresponding topological objects, Hurwitz maps, are of high genus. We prove that the limiting behaviour exhibits a new, twofold, phenomenon: the first part becomes very large, while the rest of the partition has the standard Vershik-Kerov-Logan-Shepp limit shape. As a consequence, we obtain asymptotic estimates for unconnected Hurwitz numbers with linear Euler characteristic, which we use to study random Hurwitz maps in this regime. This result can also be interpreted as the return probability of the transposition random walk on the symmetric group after linearly many steps.