  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, 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--Erd\H{o}s--Schr\"oder of eigenvalue
gaps of GUE; and the use of the theory of determinantal processes to analyze
the GUE minor process.
