# A recursion formula for the moments of the gaussian orthogonal ensemble

Annales de l'I.H.P. Probabilités et statistiques (2009)

- Volume: 45, Issue: 3, page 754-769
- ISSN: 0246-0203

## Access Full Article

top## Abstract

top## How to cite

topLedoux, M.. "A recursion formula for the moments of the gaussian orthogonal ensemble." Annales de l'I.H.P. Probabilités et statistiques 45.3 (2009): 754-769. <http://eudml.org/doc/78042>.

@article{Ledoux2009,

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.},

author = {Ledoux, M.},

journal = {Annales de l'I.H.P. Probabilités et statistiques},

keywords = {gaussian orthogonal ensemble; moment recursion formula; map enumeration; largest eigenvalue; small deviation inequality; Gaussian orthogonal ensembles; Gaussian symplectic ensembles; five-term-recurrence formula for moments; Wigner semicircle law},

language = {eng},

number = {3},

pages = {754-769},

publisher = {Gauthier-Villars},

title = {A recursion formula for the moments of the gaussian orthogonal ensemble},

url = {http://eudml.org/doc/78042},

volume = {45},

year = {2009},

}

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

PB - Gauthier-Villars

VL - 45

IS - 3

SP - 754

EP - 769

AB - 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.

LA - eng

KW - gaussian orthogonal ensemble; moment recursion formula; map enumeration; largest eigenvalue; small deviation inequality; Gaussian orthogonal ensembles; Gaussian symplectic ensembles; five-term-recurrence formula for moments; Wigner semicircle law

UR - http://eudml.org/doc/78042

ER -

## References

top- [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. Zbl1070.15013MR2126983
- [2] W. Bryc and V. Pierce. Duality of real and quaternionic random matrices, 2008. Zbl1188.15034MR2480549
- [3] L. Caffarelli. Monotonicity properties of optimal transportation and the FKG and related inequalities. Comm. Math. Phys. 214 (2000) 547–563. Zbl0978.60107MR1800860
- [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. Zbl0997.47033MR1677884
- [5] I. Goulden and D. Jackson. Maps in locally orientable surfaces and integrals over real symmetric surfaces. Can. J. Math. 49 (1997) 865–882. Zbl0903.05016MR1604106
- [6] U. Haagerup and S. Thorbjørnsen. Random matrices with complex Gaussian entries. Expo. Math. 21 (2003) 293–337. Zbl1041.15018MR2022002
- [7] J. Harer and D. Zagier. The Euler characteristic of the moduli space of curves. Invent. Math. 85 (1986) 457–485. Zbl0616.14017MR848681
- [8] W. König. Orthogonal polynomial ensembles in probability theory. Probab. Surv. 2 (2005) 385–447. Zbl1189.60024MR2203677
- [9] B. Lass. Démonstration combinatoire de la formule de Harer–Zagier. C. R. Acad. Sci. Paris Ser. I Math. 333 (2001) 155–160. Zbl0984.05006MR1851616
- [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. Zbl1045.15012MR2053053
- [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. Zbl1073.60037MR2041832
- [12] M. L. Mehta. Random Matrices. Academic Press, Boston, MA, 1991. Zbl0780.60014MR1083764
- [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. Zbl1033.81062MR2005857
- [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. Zbl1130.82313MR2191882
- [16] H. Schultz. Non-commutative polynomials of independent Gaussian random matrices. The real and symplectic cases. Probab. Theory Related Fields 131 (2005) 261–309. Zbl1085.46045MR2117954
- [17] A. Soshnikov. Universality at the edge of the spectrum in Wigner random matrices. Comm. Math. Phys. 207 (1999) 697–733. Zbl1062.82502MR1727234
- [18] G. Szegö. Orthogonal Polynomials. Colloquium Publications XXIII. Amer. Math. Soc., Providence, RI, 1975. Zbl0305.42011MR372517JFM65.0278.03
- [19] C. Tracy and H. Widom. Level-spacing distribution and the Airy kernel. Comm. Math. Phys. 159 (1994) 151–174. Zbl0789.35152MR1257246
- [20] C. Tracy and H. Widom. On orthogonal and symplectic matrix ensembles. Comm. Math. Phys. 177 (1996) 727–754. Zbl0851.60101MR1385083
- [21] E. Wigner. Characteristic vectors of bordered matrices with infinite dimensions. Ann. Math. 62 (1955) 548–564. Zbl0067.08403MR77805
- [22] A. Zvonkin. Matrix integrals and map enumeration: An accessible introduction. Math. Comput. Modelling 26 (1997) 281–304. Zbl1185.81083MR1492512

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.