Extremal laws for the real Ginibre ensemble

Abstract

The real Ginibre ensemble refers to the family of $n\times n$ matrices in which each entry is an independent Gaussian random variable of mean zero and variance one. Our main result is that the appropriately scaled spectral radius converges in law to a Gumbel distribution as $n\to \infty$. This fact has been known to hold in the complex and quaternion analogues of the ensemble for some time, with simpler proofs. Along the way we establish a new form for the limit law of the largest real eigenvalue.

Published in at http://dx.doi.org/10.1214/13-AAP958 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (3)

• General Eigenvalue Correlations for the Real Ginibre Ensemble
• A method to calculate correlation functions for $\beta =1$ random matrices of odd size
• A Note on the Pfaffian Integration Theorem

Cited by in corpus (13)

• Extreme statistics and index distribution in the classical $1d$ Coulomb gas
• Spectral radius of random matrices with independent entries
• A Note on Mixed Matrix Moments for the Complex Ginibre Ensemble
• Statistics of the maximal distance and momentum in a trapped Fermi gas at low temperature
• From Painlevé to Zakharov-Shabat and beyond: Fredholm determinants and integro-differential hierarchies
• Tilted elastic lines with columnar and point disorder, non-Hermitian quantum mechanics and spiked random matrices: pinning and localization
• Directional Extremal Statistics for Ginibre Eigenvalues
• Asymptotic expansions for a class of Fredholm Pfaffians and interacting particle systems
• The asymptotic distribution of the condition number for random circulant matrices
• Examples of interacting particle systems on $\mathbb{Z}$ as Pfaffian point processes: coalescing branching random walks and annihilating random walks with immigration
• Non-Randomness of Google's Quantum Supremacy Benchmark
• Optimal system size for complex dynamics in random neural networks near criticality
• Sharp asymptotics for Fredholm Pfaffians related to interacting particle systems and random matrices