Finite N precursors of the free cumulants

• Sylvain Lacroix (LPTHE, Sorbonne Université)

In this talk, I will discuss various properties of a family of finite N precursors of the free cumulants. They are polynomials on the space of NxN matrices, invariant under conjugations and extracted from the so-called HCIZ integral. Their main feature is their additivity with respect to the average over sums of U(N)-conjugacy orbits. Moreover, they admit natural expansions in terms of Newton and Schur polynomials, related to the representation theory of the unitary and symmetric groups and recovering a previous construction by Capitaine and Casalis. I will further discuss their large N limit, which yields the free cumulants, with 1/N² corrections expressed in terms of Hurwitz numbers. Finally, expanding on the main additivity property of these objects, I will describe the behaviour of general U(N)-invariant polynomials under sums of conjugacy orbits. This talk is based on arXiv:2508.21483, in collaboration with Jean-Bernard Zuber.

Random Lozenge Waterfall: Dimensional Collapse of Gibbs Measures

• Alisa Knizel (Barnard College, Columbia University)

I will discuss a two-parameter family of random lozenge tilings related to the q-Racah orthogonal polynomials. The model is a generalization of the uniform and q-weighted random tilings and due to the presence of additional parameters it displays a new interesting behavior. In particular, I will explain a new phenomenon that can be observed in the so-called waterfall region, where two-dimensional pure states collapse into one dimension. This is based on a joint work with L. Petrov.