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

Abstract

Associated to two given sequences of eigenvalues \(\lambda _1 \ge \cdots \ge \lambda _n\) and \(\mu _1 \ge \cdots \ge \mu _n\) 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 \(n \rightarrow \infty \). 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.