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Random Partitions Under the Plancherel-Hurwitz Measure, High Genus Hurwitz Numbers and Maps

Authors Guillaume Chapuy , Baptiste Louf , Harriet Walsh



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Author Details

Guillaume Chapuy
  • Université Paris Cité, IRIF, CNRS, F-75013 Paris, France
Baptiste Louf
  • Department of Mathematics, Uppsala University, PO Box 480, SE-751 06 Uppsala, Sweden
Harriet Walsh
  • Univ Lyon, ENS de Lyon, CNRS, Laboratoire de Physique, F-69342 Lyon, France
  • Université Paris Cité, IRIF, CNRS, F-75013 Paris, France

Acknowledgements

We thank Philippe Biane, Jérémie Bouttier and Andrea Sportiello for insightful conversations.

Cite AsGet BibTex

Guillaume Chapuy, Baptiste Louf, and Harriet Walsh. Random Partitions Under the Plancherel-Hurwitz Measure, High Genus Hurwitz Numbers and Maps. In 33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 225, pp. 6:1-6:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.AofA.2022.6

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. It appears naturally in the enumeration of Hurwitz maps, or equivalently 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 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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Enumeration
  • Mathematics of computing → Distribution functions
  • Mathematics of computing → Random graphs
Keywords
  • Random partitions
  • limit shapes
  • transposition factorisations
  • map enumeration
  • Hurwitz numbers
  • RSK algorithm
  • giant components

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