Random Hermitian Matrices and Gaussian Multiplicative Chaos
Probability
2017-09-19 v4 Mathematical Physics
math.MP
Abstract
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 -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.
Cite
@article{arxiv.1701.03289,
title = {Random Hermitian Matrices and Gaussian Multiplicative Chaos},
author = {Nathanaël Berestycki and Christian Webb and Mo Dick Wong},
journal= {arXiv preprint arXiv:1701.03289},
year = {2017}
}
Comments
Version 3: analysis of the differential identities simplified slightly. Version 4: some errors fixed