Elnur Emrah: Boundary fluctuations for uniform Gelfand–Tsetlin pattern with fixed top level

Abstract:

A Gelfand–Tsetlin (GT) of depth \(n\) is an interlacing array of real entries distributed over \(n\) levels such that level \(k = 0, 1, \dotsc, n-1\) contains exactly \(n-k\) entries. This object can be identified with the spectra of the principal submatrices of an \(n \times n\) 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 (in probability) for extremal entries, which follows as a consequence.