eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-06-08
6:1
6:12
10.4230/LIPIcs.AofA.2022.6
article
Random Partitions Under the Plancherel-Hurwitz Measure, High Genus Hurwitz Numbers and Maps
Chapuy, Guillaume
1
https://orcid.org/0000-0001-7730-5417
Louf, Baptiste
2
https://orcid.org/0000-0002-7799-6948
Walsh, Harriet
3
1
https://orcid.org/0000-0002-4199-4975
Université Paris Cité, IRIF, CNRS, F-75013 Paris, France
Department of Mathematics, Uppsala University, PO Box 480, SE-751 06 Uppsala, Sweden
Univ Lyon, ENS de Lyon, CNRS, Laboratoire de Physique, F-69342 Lyon, France
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol225-aofa2022/LIPIcs.AofA.2022.6/LIPIcs.AofA.2022.6.pdf
Random partitions
limit shapes
transposition factorisations
map enumeration
Hurwitz numbers
RSK algorithm
giant components