  The goal of this article is to study how much the eigenvalues of large
Hermitian random matrices deviate from certain deterministic locations -- or in
other words, to investigate optimal rigidity estimates for the eigenvalues. We
do this in the setting of one-cut regular unitary invariant ensembles of random
Hermitian matrices -- the Gaussian Unitary Ensemble being the prime example of
such an ensemble. Our approach to this question combines extreme value theory
of log-correlated stochastic processes, and in particular the theory of
multiplicative chaos, with asymptotic analysis of large Hankel determinants
with Fisher-Hartwig symbols of various types, such as merging jump
singularities, size-dependent impurities, and jump singularities approaching
the edge of the spectrum. In addition to optimal rigidity estimates, our
approach sheds light on the fractal geometry of the eigenvalue counting
function.
