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Fluctuations and correlations for products of real asymmetric random matrices

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posted on 2023-06-10, 04:48 authored by Will FitzGerald, Nicholas SimmNicholas Simm
We study the real eigenvalue statistics of products of independent real Ginibre random matrices. These are matrices all of whose entries are real i.i.d. standard Gaussian random variables. For such product ensembles, we demonstrate the asymptotic normality of suitably normalised linear statistics of the real eigenvalues and compute the limiting variance explicitly in both global and mesoscopic regimes. A key part of our proof establishes uniform decorrelation estimates for the related Pfaffian point process, thereby allowing us to exploit weak dependence of the real eigenvalues to give simple and quick proofs of the central limit theorems under quite general conditions. We also establish the universality of these point processes. We compute the asymptotic limit of all correlation functions of the real eigenvalues in the bulk, origin and spectral edge regimes. By a suitable strengthening of the convergence at the edge, we also obtain the limiting fluctuations of the largest real eigenvalue. Near the origin we find new limiting distributions characterising the smallest positive real eigenvalue.

History

Publication status

  • Published

File Version

  • Accepted version

Journal

L'Institut Henri Poincare, Annales B: Probabilites et Statistiques

ISSN

0246-0203

Publisher

Institute of Mathematical Statistics

Issue

4

Volume

59

Page range

2308-2342

Department affiliated with

  • Mathematics Publications

Institution

University of Sussex

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2022-09-23

First Open Access (FOA) Date

2022-09-23

First Compliant Deposit (FCD) Date

2022-09-22

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