### CLASSIFYING BOUNDARY FLUCTUATIONS FOR UNIFORMLY RANDOM GELFAND-TSETLIN PATTERNS

Probability Seminar

28th April 2023, 3:30 pm – 4:30 pm

Fry Building, 2.04

A Gelfand-Tsetlin (GT) pattern of depth n ∈ Z_{>0} is an interlacing array of 1/2n(n+1) real entries distributed over n levels such that level k ∈ {0, 1, . . . , n−1} contains exactly n − k entries. This object naturally arises from an n × n Hermitian matrix by placing the eigenvalues of the (n − k) × (n − k) leading principal submatrix on level k. Let G_n denote a GT pattern of depth n with a fixed increasing sequence a_n ∈ R^n on level 0 and with the remaining entries viewed as particles chosen uniformly at random. In this talk, our interest is in the limiting boundary fluctuations of G_n at the level of finite-dimensional distributions of first particles. We present a classification theorem that identifies five fluctuation regimes in terms of the level zero data {a_n : n ∈ Z_{>0}} and describes the corresponding limit processes. This result is from a forthcoming joint work with Kurt Johansson

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