Sums of GUE matrices and concentration of hives from correlation decay of eigengaps

2023/06/20 by Narayanan, Hariharan, Sheffield, Scott, Tao, Terence
#60B20 #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2306.11514

Abstract

Associated to two given sequences of eigenvalues λ₁ ≥ … ≥ λ_n and μ₁ ≥ … ≥ μ_n is a natural polytope, the polytope of augmented hives with the specified boundary data, which is associated to sums of random Hermitian matrices with these eigenvalues. As a first step towards the asymptotic analysis of random hives, we show that if the eigenvalues are drawn from the GUE ensemble, then the associated augmented hives exhibit concentration as n → ∞. Our main ingredients include a representation due to Speyer of augmented hives involving a supremum of linear functions applied to a product of Gelfand--Tsetlin polytopes; known results by Klartag on the KLS conjecture in order to handle the aforementioned supremum; covariance bounds of Cipolloni--Erdős--Schröder of eigenvalue gaps of GUE; and the use of the theory of determinantal processes to analyze the GUE minor process.

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