Soutenance de Harriet Walsh
| When |
Oct 26, 2022
from 03:00 to 05:00 |
|---|---|
| Where | Salle des thèses |
| Contact Name | Harriet Walsh |
| Attendees |
Harriet Walsh |
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Random integer partitions have been particularly useful in explaining the connections between diverse physical and combinatorial models exhibiting the same asymptotic phenomena. Famously, a partition under the Plancherel measure encodes the lengths of monotone subsequences of a uniform random permutation; its parts also correspond to positions of free fermions on a lattice, a connection that allows their statistics to be studied exactly. It has a deterministic limit shape and edge fluctuations with a universal critical exponent of 1/3, associated with out-of-equilibrium physics. This thesis presents two generalisations of the Plancherel measure with edge behaviour escaping its universality class. First, we introduce measures on partitions corresponding to natural models of free fermions, and show that they give rise to “multicritical” asymptotic edge fluctuations, with new critical exponents. These measures relate multicritical free fermions to random unitary matrices, explaining the appearance of the same asymptotic distributions for both. Second, we introduce a measure related to the enumeration of transposition factorisations on symmetric groups and certain discrete surfaces. We show that, in a regime where the corresponding surfaces are of high genus, it produces a novel twofold limit behaviour where the first part becomes very large. As a consequence, we find an asymptotic estimate for the unconnected Hurwitz numbers at high genus. The laws studied each have integrable structures. In the first case our analysis exploits integrability directly; in the second, we use an entropy method to study an asymptotic regime which is inaccessible by integrability approaches.