Topological expansion of unitary integrals and maps

• Thomas Buc d'Alché (ENS de Lyon)

The problem of computing the large N asymptotics of multi-matrix models may be reduced in many cases to the computation of integrals over the unitary group. We explain how these asympotics can be computed (in the perturbative regime) using Weingarten calculus and Dyson-Schwinger equations. The terms of the asymptotic expansion are expressed in terms of maps, i.e. graphs embedded in surfaces. The particular maps involved generalize the monotone Hurwitz numbers. This is an example of the rich links between combinatorics and random matrix theory, which originated in quantum field theory.

Asymptotic behaviour of solution to the KdV hierarchy in deformed unitary ensembles

• Carla Mariana da Silva Pinheiro (Universidad Sao Carlos)

The eigenvalues in random matrix ensembles model several systems, such as g-maps enumeration, big data, quantum chaos, and even the distribution of trees in a forest. In unitary ensembles, eigenvalues constitute a determinantal point process, whose statistics are encoded in a kernel. We study a deformation of the kernel that arises when the density of the equilibrium measure vanishes like a power of 5/2. We show that the statistics in this ensemble relate to the three first equations of the KdV hierarchy. Moreover, we derive the asymptotics for such solutions. This is based on a work in collaboration with Mattia Cafasso (arXiv:2504.20721).