## Electronic Journal of Probability

### Differential Operators and Spectral Distributions of Invariant Ensembles from the Classical Orthogonal Polynomials. The Continuous Case

Michel Ledoux

#### Abstract

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.

#### Article information

Source
Electron. J. Probab., Volume 9 (2004), paper no. 7, 177-208.

Dates
Accepted: 15 February 2004
First available in Project Euclid: 6 June 2016

https://projecteuclid.org/euclid.ejp/1465229693

Digital Object Identifier
doi:10.1214/EJP.v9-191

Mathematical Reviews number (MathSciNet)
MR2041832

Zentralblatt MATH identifier
1073.60037

Rights

#### Citation

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

#### References

• Aubrun, G. An inequality about the largest eigenvalue of a random matrix (2003). To appear in Séminaire de Probabilités XXXVIII. Lecture Notes in Math., Springer.
• Bai, Z. D. Methodologies in spectral analysis of large-dimensional random matrices, a review. With comments by G. J. Rodgers and Jack W. Silverstein; and a rejoinder by the author. Statist. Sinica 9 (1999), no. 3, 611–677.
• Baker, T. H.; Forrester, P. J.; Pearce, P. A. Random matrix ensembles with an effective extensive external charge. J. Phys. A 31 (1998), no. 29, 6087–6101.
• Bakry, Dominique. L'hypercontractivité et son utilisation en théorie des semigroupes. (French) [Hypercontractivity and its use in semigroup theory] Lectures on probability theory (Saint-Flour, 1992), 1–114, Lecture Notes in Math., 1581, Springer, Berlin, 1994.
• Ben Arous, G.; Dembo, A.; Guionnet, A. Aging of spherical spin glasses. Probab. Theory Related Fields 120 (2001), no. 1, 1–67.
• Capitaine, M.; Casalis, M. Asymptotic freeness by generalized moments for Gaussian and Wishart matrices. Application to Beta random matrices (2002). To appear in Indiana J. Math.
• Chen, Li-Chen; Ismail, Mourad E. H. On asymptotics of Jacobi polynomials. SIAM J. Math. Anal. 22 (1991), no. 5, 1442–1449.
• Chihara, T. S. An introduction to orthogonal polynomials. Mathematics and its Applications, Vol. 13. Gordon and Breach Science Publishers, New York-London-Paris, 1978. xii+249 pp. ISBN: 0-677-04150-0.
• Collins, B. Product of random projections, Jacobi ensembles and universality problems arising from free probability (2003).
• Deift, P. A. Orthogonal polynomials and random matrices: a Riemann-Hilbert approach. Courant Lecture Notes in Mathematics, 3. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999. viii+273 pp. ISBN: 0-9658703-2-4; 0-8218-2695-6.
• Forrester, P.J. Log-gases and random matrices (2002).
• Fukushima, Masatoshi; Oshima, Yoichi; Takeda, Masayoshi. Dirichlet forms and symmetric Markov processes. de Gruyter Studies in Mathematics, 19. Walter de Gruyter & Co., Berlin, 1994. x+392 pp. ISBN: 3-11-011626-X.
• Grenander, Ulf; Silverstein, Jack W. Spectral analysis of networks with random topologies. SIAM J. Appl. Math. 32 (1977), no. 2, 499–519.
• Guionnet, A.; Zeitouni, O. Concentration of the spectral measure for large matrices. Electron. Comm. Probab. 5 (2000), 119–136.
• Haagerup, U.; Thorbjørnsen, S. Random matrices with complex Gaussian entries. (1997).
• Harer, J.; Zagier, D. The Euler characteristic of the moduli space of curves. Invent. Math. 85 (1986), no. 3, 457–485.
• R. Koekoek, R.; Swarttouw, R. F. The Askey scheme of hypergeometric orthogonal polynomials and its q-analogue (1998). http://aw.twi.tudelft.nl/~koekoek/askey/index.html
• Kuijlaars, A. B. J.; Van Assche, W. The asymptotic zero distribution of orthogonal polynomials with varying recurrence coefficients. J. Approx. Theory 99 (1999), no. 1, 167–197.
• Johansson, Kurt. Shape fluctuations and random matrices. Comm. Math. Phys. 209 (2000), no. 2, 437–476.
• Johansson, Kurt. Discrete orthogonal polynomial ensembles and the Plancherel measure. Ann. of Math. (2) 153 (2001), no. 1, 259–296.
• Johnstone, Iain M. On the distribution of the largest eigenvalue in principal components analysis. Ann. Statist. 29 (2001), no. 2, 295–327.
• Jonsson, Dag. Some limit theorems for the eigenvalues of a sample covariance matrix. J. Multivariate Anal. 12 (1982), no. 1, 1–38.
• Ledoux, M. A remark on hypercontractivity and tail inequalities for the largest eigenvalues of random matrices. Séminaire de Probabilités XXXVI. [Seminar on Probability Theory XXXVI], p.360-369. Edited by J. Azéma, M. Émery, M. Ledoux and M. Yor. Lecture Notes in Mathematics, 1801. Springer-Verlag, Berlin, 2003. viii+497 pp. ISBN: 3-540-00072-0.
• Ledoux, M. Differential operators and spectral distributions of invariant ensembles from the classical orthogonal polynomials. The discrete case. (2003).
• Marčenko, V. A.; Pastur, L. A. Distribution of eigenvalues in certain sets of random matrices. (Russian) Mat. Sb. (N.S.) 72 (114) 1967 507–536.
• Máté, Attila; Nevai, Paul; Totik, Vilmos. Strong and weak convergence of orthogonal polynomials. Amer. J. Math. 109 (1987), no. 2, 239–281.
• Mazet, Olivier. Classification des semi-groupes de diffusion sur $\mathbb{R}$ associés à une famille de polynômes orthogonaux. (French) [Classification of diffusion semigroups in $\mathbb{R}$ associated with orthogonal polynomials] Séminaire de Probabilités, XXXI, 40–53, Lecture Notes in Math., 1655, Springer, Berlin, 1997.
• Mehta, M. L. Random matrices. Academic Press (1991).
• Szegö, Gábor. Orthogonal polynomials. Fourth edition. American Mathematical Society, Colloquium Publications, Vol. XXIII. American Mathematical Society, Providence, R.I., 1975. xiii+432 pp.
• Tracy, Craig A.; Widom, Harold. Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 (1994), no. 1, 151–174.
• Tracy, Craig A.; Widom, Harold. Level spacing distributions and the Bessel kernel. Comm. Math. Phys. 161 (1994), no. 2, 289–309.
• Voiculescu, Dan. A strengthened asymptotic freeness result for random matrices with applications to free entropy. Internat. Math. Res. Notices 1998, no. 1, 41–63.
• Wachter, Kenneth W. The strong limits of random matrix spectra for sample matrices of independent elements. Ann. Probability 6 (1978), no. 1, 1–18.
• Wigner, Eugene P. Characteristic vectors of bordered matrices with infinite dimensions. Ann. of Math. (2) 62 (1955), 548–564.