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

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

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## Abstract

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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.

## How to cite

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Ledoux, 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

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