Random Hermitian matrices and Gaussian multiplicative chaos

Nathanaël Berestycki; Christian Webb; Mo Dick Wong

doi:10.1007/s00440-017-0806-9

Random Hermitian matrices and Gaussian multiplicative chaos

Nathanaël Berestycki^*, Christian Webb, Mo Dick Wong

^*Corresponding author for this work

Department of Mathematics and Systems Analysis

Research output: Contribution to journal › Article › Scientific › peer-review

14 Citations (Scopus)

113 Downloads (Pure)

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 (Formula presented.)-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.

Original language	English
Pages (from-to)	103-189
Number of pages	87
Journal	Probability Theory and Related Fields
Volume	172
Issue number	1-2
Early online date	6 Nov 2017
DOIs	https://doi.org/10.1007/s00440-017-0806-9
Publication status	Published - Oct 2018
MoE publication type	A1 Journal article-refereed

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10.1007/s00440-017-0806-9

Berestycki2018_Article_RandomHermitianMatricesAndGausFinal published version, 1.35 MBLicence: CC BY

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2 Finished

Random geometry in number theory, combinatorics, and random matrix theory
Webb, C.
01/09/2017 → 31/08/2020
Project: Academy of Finland: Other research funding
Algebraic structures and random geometry of stochastic lattice models
Kytölä, K., Webb, C., Karrila, A., Radnell, D., Gutiérrez, A. W., Kohl, F., Orlich, M. & Abuzaid, O.
01/09/2015 → 31/08/2019
Project: Academy of Finland: Other research funding

Cite this

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title = "Random Hermitian matrices and Gaussian multiplicative chaos",

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 (Formula presented.)-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.",

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AB - 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 (Formula presented.)-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.

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Random Hermitian matrices and Gaussian multiplicative chaos

Abstract

Access to Document

Random geometry in number theory, combinatorics, and random matrix theory

Algebraic structures and random geometry of stochastic lattice models

Cite this