Random matrices, Young diagrams and plane trees

• Fabio Deelan Cunden (Università degli Studi di Bari)

We consider ‘λ-shaped random matrices’, whose entries are i.i.d. in the boxes of a given Young diagram λ and zero elsewhere. In particular, we study their limiting spectral distribution when the shape λ is dilated by a growing factor N. The moments of such a distribution are a generalisation of Catalan numbers, and enumerate a combinatorial object which we call λ-plane trees: these are trees whose vertices are labelled in a way that is ‘compatible’ with λ. Based on joint works with Elia Bisi and Marilena Ligabò.

Fredholm determinants from Schrödinger-type equations, and deformation of Tracy-Widom distribution

• Taro Kimura (Université de Bourgogne)

The Tracy-Widom distribution, defined via the Fredholm determinant associated with the Airy kernel, has emerged as a universal distribution in various contexts, including statistical mechanics, stochastic processes, and related fields. The aim of this presentation is to introduce our recent approach in establishing a novel deformation of the Tracy-Widom distribution, linked to a broad class of Schrödinger-type ODE systems. I will elucidate the underlying mathematical structures of this method, including a nonlinear integro-differential equation as a deformation of the Painlevé II equation, the hierarchical Poisson structure, and its connection to the isomonodromic system. This is based on a joint-work with Xavier Navand, https://arxiv.org/abs/2408.06888