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.

29 pages, 6 figures. v2: minor corrections made and details added, in particular to Proposition 4.3 and its proof

References in corpus (8)

• A renormalisation group method. II. Approximation by local polynomials
• A renormalisation group method. III. Perturbative analysis
• A renormalisation group method. IV. Stability analysis
• A renormalisation group method. I. Gaussian integration and normed algebras
• A renormalisation group method. V. A single renormalisation group step
• An elementary proof of phase transition in the planar XY model
• The Fröhlich-Spencer Proof of the Berezinskii-Kosterlitz-Thouless Transition
• Variance of voltages in a lattice Coulomb gas