Electronic Journal of Probability
- Electron. J. Probab.
- Volume 9 (2004), paper no. 7, 177-208.
Differential Operators and Spectral Distributions of Invariant Ensembles from the Classical Orthogonal Polynomials. The Continuous Case
Following the investigation by U. Haagerup and S. Thorbjornsen, we present a simple differential approach to the limit theorems for empirical spectral distributions of complex random matrices from the Gaussian, Laguerre and Jacobi Unitary Ensembles. In the framework of abstract Markov diffusion operators, we derive by the integration by parts formula differential equations for Laplace transforms and recurrence equations for moments of eigenfunction measures. In particular, a new description of the equilibrium measures as adapted mixtures of the universal arcsine law with an independent uniform distribution is emphasized. The moment recurrence relations are used to describe sharp, non asymptotic, small deviation inequalities on the largest eigenvalues at the rate given by the Tracy-Widom asymptotics.
Electron. J. Probab., Volume 9 (2004), paper no. 7, 177-208.
Accepted: 15 February 2004
First available in Project Euclid: 6 June 2016
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60F99: None of the above, but in this section
Secondary: 33C99: None of the above, but in this section 60J25: Continuous-time Markov processes on general state spaces 60J60: Diffusion processes [See also 58J65] 82B31: Stochastic methods 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)
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Ledoux, Michel. Differential Operators and Spectral Distributions of Invariant Ensembles from the Classical Orthogonal Polynomials. The Continuous Case. Electron. J. Probab. 9 (2004), paper no. 7, 177--208. doi:10.1214/EJP.v9-191. https://projecteuclid.org/euclid.ejp/1465229693