On the distribution of eigenvalues of GUE and its minors at fixed index

We obtain bounds on the distribution of normalized gaps of eigenvalues of $N\times N$ GUE matrix in the bulk, that do not lose logarithmic factors of $N$ in the limit $N\to \mathrm{\infty }$. As an application, we obtain fixed index universality results for the GUE minor process, which in turn are useful for establishing limiting results for random hives with GUE boundary data.