EXTREME GAPS BETWEEN EIGENVALUES OF RANDOM MATRICES

• Published in: The Annals of probability Vol. 41; no. 4; pp. 2648 - 2681
• Main Authors: Ben Arous, Gérard, Bourgade, Paul
• Format: Journal Article
• Language: English
• Published: Hayward Institute of Mathematical Statistics 01-07-2013; The Institute of Mathematical Statistics
• Subjects: 11M50; 15B52; 60B20; Density; Determinants; Eigenvalues; Eigenvalues statistics; extreme spacings; Factorials; Gaussian unitary ensemble; Mathematical analysis; Mathematical functions; Mathematical theorems; Matrices; Matrix; negative association property; Normal distribution; random matrices; Random variables; Spectral theory; Studies
• ISSN: 0091-1798; 2168-894X
• DOI: 10.1214/11-AOP710

This paper studies the extreme gaps between eigenvalues of random matrices. We give the joint limiting law of the smallest gaps for Haar-distributed unitary matrices and matrices from the Gaussian unitary ensemble. In particular, the kth smallest gap, normalized by a factor n -4/3 , has a limiting density proportional to ${\mathrm{x}}^{3\mathrm{k}-1}{\mathrm{e}}^{-{\mathrm{x}}^{3}}$ . Concerning the largest gaps, normalized by n/√logn, they converge in L p to a constant for all p > 0. These results are compared with the extreme gaps between zeros of the Riemann zeta function.