Associated to two given sequences of eigenvalues $λ_1 \geq \dots \geq λ_n$ and $μ_1 \geq \dots \geq μ_n$ 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 $n \rightarrow \infty$. 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.