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Associated to two given sequences of eigenvalues $\lambda_1 \geq \dots \geq\lambda_n$ and $\mu_1 \geq \dots \geq \mu_n$ is a natural polytope, thepolytope of augmented hives with the specified boundary data, which isassociated to sums of random Hermitian matrices with these eigenvalues. As afirst step towards the asymptotic analysis of random hives, we show that if theeigenvalues are drawn from the GUE ensemble, then the associated augmentedhives exhibit concentration as $n \rightarrow \infty$. Our main ingredientsinclude a representation due to Speyer of augmented hives involving a supremumof linear functions applied to a product of Gelfand--Tsetlin polytopes; knownresults by Klartag on the KLS conjecture in order to handle the aforementionedsupremum; covariance bounds of Cipolloni--Erd\H{o}s--Schr\"oder of eigenvaluegaps of GUE; and the use of the theory of determinantal processes to analyzethe GUE minor process.

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