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.
Publié le : 2009-08-15
Classification:
Gaussian Orthogonal Ensemble,
Moment recursion formula,
Map enumeration,
Largest eigenvalue,
Small deviation inequality,
46L54,
15A52,
33C45,
60E05,
82B31
@article{1249391383,
author = {Ledoux, M.},
title = {A recursion formula for the moments of the Gaussian orthogonal ensemble},
journal = {Ann. Inst. H. Poincar\'e Probab. Statist.},
volume = {45},
number = {1},
year = {2009},
pages = { 754-769},
language = {en},
url = {http://dml.mathdoc.fr/item/1249391383}
}
Ledoux, M. A recursion formula for the moments of the Gaussian orthogonal ensemble. Ann. Inst. H. Poincaré Probab. Statist., Tome 45 (2009) no. 1, pp. 754-769. http://gdmltest.u-ga.fr/item/1249391383/