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Fluctuations and correlations for products of real asymmetric random matrices
journal contribution
posted on 2023-06-10, 04:48 authored by Will FitzGerald, Nicholas SimmNicholas SimmWe 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 StatistiquesISSN
0246-0203Publisher
Institute of Mathematical StatisticsPublisher URL
External DOI
Issue
4Volume
59Page range
2308-2342Department affiliated with
- Mathematics Publications
Institution
University of SussexFull text available
- Yes
Peer reviewed?
- Yes
Legacy Posted Date
2022-09-23First Open Access (FOA) Date
2022-09-23First Compliant Deposit (FCD) Date
2022-09-22Usage metrics
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