Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

A recursion formula for the moments of the Gaussian orthogonal ensemble

M. Ledoux

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Abstract

We present an analogue of the Harer–Zagier recursion formula for the moments of the Gaussian Orthogonal Ensemble in the form of a five term recurrence equation. The proof is based on simple Gaussian integration by parts and differential equations on Laplace transforms. A similar recursion formula holds for the Gaussian Symplectic Ensemble. As in the complex case, the result is interpreted as a recursion formula for the number of 1-vertex maps in locally orientable surfaces with a given number of edges and faces. This moment recurrence formula is also applied to a sharp bound on the tail of the largest eigenvalue of the Gaussian Orthogonal Ensemble and, by moment comparison, of families of Wigner matrices.

Résumé

Ce travail présente un analogue de la relation de récurrence de Harer et Zagier pour les moments de l’Ensemble Orthogonal Gaussien sous la forme d’une récurrence à cinq termes. La démonstration s’appuie sur des intégrations par parties gaussiennes et des équations différentielles sur les transformées de Laplace. Une relation similaire est établie pour l’Ensemble Symplectique Gaussien. Comme dans le cas complexe, cette relation s’interprète comme une formule de récurrence pour le nombre de cartes enracinées à nombre de faces et de côtés donné plongées dans des surfaces localement orientées. Cette relation de récurrence sur les moments fournit également une borne sur la loi de la plus grande valeur propre de l’Ensemble Orthogonal Gaussien et, par comparaison de moments, de familles de matrices de Wigner.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 45, Number 3 (2009), 754-769.

Dates
First available in Project Euclid: 4 August 2009

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1249391383

Digital Object Identifier
doi:10.1214/08-AIHP184

Mathematical Reviews number (MathSciNet)
MR2548502

Zentralblatt MATH identifier
1184.60003

Subjects
Primary: 46L54: Free probability and free operator algebras 15A52 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions] 60E05: Distributions: general theory 82B31: Stochastic methods

Keywords
Gaussian Orthogonal Ensemble Moment recursion formula Map enumeration Largest eigenvalue Small deviation inequality

Citation

Ledoux, M. A recursion formula for the moments of the Gaussian orthogonal ensemble. Ann. Inst. H. Poincaré Probab. Statist. 45 (2009), no. 3, 754--769. doi:10.1214/08-AIHP184. https://projecteuclid.org/euclid.aihp/1249391383


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