Random partitions under the Plancherel-Hurwitz measure, high genus Hurwitz numbers and maps

TL;DR

This paper introduces the Plancherel-Hurwitz measure to analyze the asymptotic behavior of random partitions, revealing a new phenomenon in high genus Hurwitz maps and providing asymptotic estimates for related combinatorial quantities.

Contribution

It presents a novel probability measure on partitions linked to Hurwitz numbers, exploring its asymptotics in high genus regimes and connecting to random walks on symmetric groups.

Findings

Partition parts exhibit a twofold asymptotic behavior.

The limit shape aligns with the Vershik-Kerov-Logan-Shepp profile for most of the partition.

Asymptotic estimates for unconnected Hurwitz numbers with linear Euler characteristic.

Abstract

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.