2019/06/04 by Claeys, Tom, Fahs, Benjamin, Lambert, Gaultier +1 · 1 citation
#47B35 #60G15 #60G57 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Primary 60B20. Secondary 35Q15 #Probability (math.PR)
paper · doi:10.48550/arxiv.1906.01561
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.