A recursion formula for the moments of the gaussian orthogonal ensemble
Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 3, pp. 754-769.

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.

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.

DOI : 10.1214/08-AIHP184
Classification : 46L54, 15A52, 33C45, 60E05, 82B31
Mots-clés : gaussian orthogonal ensemble, moment recursion formula, map enumeration, largest eigenvalue, small deviation inequality 
@article{AIHPB_2009__45_3_754_0,
     author = {Ledoux, M.},
     title = {A recursion formula for the moments of the gaussian orthogonal ensemble},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {754--769},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {3},
     year = {2009},
     doi = {10.1214/08-AIHP184},
     mrnumber = {2548502},
     zbl = {1184.60003},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/08-AIHP184/}
}
TY  - JOUR
AU  - Ledoux, M.
TI  - A recursion formula for the moments of the gaussian orthogonal ensemble
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2009
SP  - 754
EP  - 769
VL  - 45
IS  - 3
PB  - Gauthier-Villars
UR  - http://www.numdam.org/articles/10.1214/08-AIHP184/
DO  - 10.1214/08-AIHP184
LA  - en
ID  - AIHPB_2009__45_3_754_0
ER  - 
%0 Journal Article
%A Ledoux, M.
%T A recursion formula for the moments of the gaussian orthogonal ensemble
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2009
%P 754-769
%V 45
%N 3
%I Gauthier-Villars
%U http://www.numdam.org/articles/10.1214/08-AIHP184/
%R 10.1214/08-AIHP184
%G en
%F AIHPB_2009__45_3_754_0
Ledoux, M. A recursion formula for the moments of the gaussian orthogonal ensemble. Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 3, pp. 754-769. doi : 10.1214/08-AIHP184. http://www.numdam.org/articles/10.1214/08-AIHP184/

[1] G. Aubrun. An inequality about the largest eigenvalue of a random matrix. In Séminaire de Probabilités XXXVIII 320-337. Lecture Notes in Math. 1857. Springer, Berlin, 2005. | MR | Zbl

[2] W. Bryc and V. Pierce. Duality of real and quaternionic random matrices, 2008. | MR

[3] L. Caffarelli. Monotonicity properties of optimal transportation and the FKG and related inequalities. Comm. Math. Phys. 214 (2000) 547-563. | MR | Zbl

[4] P. A. Deift. Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. CIMS Lecture Notes 3. Courant Institute of Mathematical Sciences, New York, 1999. | MR | Zbl

[5] I. Goulden and D. Jackson. Maps in locally orientable surfaces and integrals over real symmetric surfaces. Can. J. Math. 49 (1997) 865-882. | MR | Zbl

[6] U. Haagerup and S. Thorbjørnsen. Random matrices with complex Gaussian entries. Expo. Math. 21 (2003) 293-337. | MR | Zbl

[7] J. Harer and D. Zagier. The Euler characteristic of the moduli space of curves. Invent. Math. 85 (1986) 457-485. | MR | Zbl

[8] W. König. Orthogonal polynomial ensembles in probability theory. Probab. Surv. 2 (2005) 385-447. | MR

[9] B. Lass. Démonstration combinatoire de la formule de Harer-Zagier. C. R. Acad. Sci. Paris Ser. I Math. 333 (2001) 155-160. | MR | Zbl

[10] M. Ledoux. A remark on hypercontractivity and tail inequalities for the largest eigenvalues of random matrices. In Séminaire de Probabilités XXXVII 360-369. Lecture Notes in Mathematics 1832. Springer, Berlin, 2003. | MR | Zbl

[11] M. Ledoux. Differential operators and spectral distributions of invariant ensembles from the classical orthogonal polynomials. The continuous case. Electron. J. Probab. 9 (2004) 177-208. | EuDML | MR | Zbl

[12] M. L. Mehta. Random Matrices. Academic Press, Boston, MA, 1991. | MR | Zbl

[13] M. Mulase and A. Waldron. Duality of orthogonal and symplectic random matrix integrals and quaternionic Feynman graphs. Comm. Math. Phys. 240 (2003) 553-586. | MR | Zbl

[14] V. Pierce. An algorithm for map enumeration (2006).

[15] A. Ruzmaikina. Universality of the edge distribution of eigenvalues of Wigner random matrices with polynomially decaying distributions of entries. Comm. Math. Phys. 261 (2006) 277-296. | MR | Zbl

[16] H. Schultz. Non-commutative polynomials of independent Gaussian random matrices. The real and symplectic cases. Probab. Theory Related Fields 131 (2005) 261-309. | MR | Zbl

[17] A. Soshnikov. Universality at the edge of the spectrum in Wigner random matrices. Comm. Math. Phys. 207 (1999) 697-733. | MR | Zbl

[18] G. Szegö. Orthogonal Polynomials. Colloquium Publications XXIII. Amer. Math. Soc., Providence, RI, 1975. | JFM | MR | Zbl

[19] C. Tracy and H. Widom. Level-spacing distribution and the Airy kernel. Comm. Math. Phys. 159 (1994) 151-174. | MR | Zbl

[20] C. Tracy and H. Widom. On orthogonal and symplectic matrix ensembles. Comm. Math. Phys. 177 (1996) 727-754. | MR | Zbl

[21] E. Wigner. Characteristic vectors of bordered matrices with infinite dimensions. Ann. Math. 62 (1955) 548-564. | MR | Zbl

[22] A. Zvonkin. Matrix integrals and map enumeration: An accessible introduction. Math. Comput. Modelling 26 (1997) 281-304. | MR | Zbl

Cité par Sources :