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

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.
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https://hal.archives-ouvertes.fr/hal-03704671
Contributor : Harriet Walsh Connect in order to contact the contributor
Submitted on : Saturday, June 25, 2022 - 1:43:10 PM
Last modification on : Friday, August 5, 2022 - 12:00:32 PM

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  • HAL Id : hal-03704671, version 1
  • ARXIV : 2206.11315

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Guillaume Chapuy, Baptiste Louf, Harriet Walsh. Random partitions under the Plancherel-Hurwitz measure, high genus Hurwitz numbers and maps. 2022. ⟨hal-03704671⟩

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