Elnur Emrah

Boundary fluctuations for uniform Gelfand--Tsetlin pattern with fixed top level

A Gelfand--Tsetlin (GT) of depth n is an interlacing array of real entries distributed over n levels such that level k = 0, 1, …, n-1 contains exactly n-k entries. This object can be identified with the spectra of the principal submatrices of an n by n Hermitian matrix. Consider now a random GT pattern produced by fixing the entries on level 0 and sampling the rest of the pattern uniformly. Through a result of Baryshnikov, an equivalent model is the eigenvalue minor process of an n by n unitarily invariant random Hermitian matrix with a fixed spectrum. In a forthcoming joint work with Kurt Johansson, we describe the multi-level fluctuations of extremal entries and identify five limit regimes. The focus of this talk is the one-level case of our main result for large levels. Then we encounter four types of limit distributions: Tracy--Widom GUE, certain generalizations of the Gaussian and Baik--Ben Arous--Péché distributions, and another novel distribution. We present this result along with a shape theorem for extremal entries, which follows as a consequence.