Random Hermitian Matrices and Gaussian Multiplicative Chaos

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Berestycki, Nathanaël 
Webb, Christian 
Wong, Mo Dick 

We prove that when suitably normalized, small enough powers of the absolute value of the characteristic polynomial of random Hermitian matrices, drawn from one-cut regular unitary invariant ensembles, converge in law to Gaussian multiplicative chaos measures. We prove this in the so-called L2-phase of multiplicative chaos. Our main tools are asymptotics of Hankel determinants with Fisher–Hartwig singularities. Using Riemann–Hilbert methods, we prove a rather general Fisher–Hartwig formula for one-cut regular unitary invariant ensembles.

math.PR, math.PR, math-ph, math.MP
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Probability Theory and Related Fields
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Springer Nature
Engineering and Physical Sciences Research Council (EP/L018896/1)
Engineering and Physical Sciences Research Council (EP/I03372X/1)
Engineering and Physical Sciences Research Council (EP/L016516/1)
N. Berestycki’s work is supported by EPSRC Grants EP/L018896/1 and EP/I03372X/1. M. D. Wong is a PhD student at the Cambridge Centre for Analysis, supported by EPSRC Grant EP/L016516/1. Some of this work was carried out while the first and third authors visited the University of Helsinki, funded in part by EPSRC Grant EP/L018896/1. They also wish to thank the University of Helsinki for its hospitality during this visit. C.Webb wishes to thank the Isaac Newton Institute for Mathematical Sciences for its hospitality during the Random Geometry program, during which this project was initiated. C.Webb was supported by the Eemil Aaltonen Foundation grant Stochastic dynamics on large random graphs and Academy of Finland Grants 288318 and 308123.