[La mesure spectrale limite du modèle de matrices aléatoires de la loi Gaussienne Inverse Généralisée]
Dans cette Note, nous explicitons la mesure spectrale limite du modèle de matrices aléatoires soumises à la loi Gaussienne Inverse Généralisée. La stratégie sous-jacente repose sur un théorème de grandes déviations établi pour la mesure spectrale de certains gaz de Coulomb plus généraux. Cette limite apparaît comme la mesure extrémale d'un problème d'énergie logarithmique avec poids qui peut être complètement résolu ici. De plus, nous prouvons la convergence presque sûre de la plus grande (resp. petite) valeur propre vers le bord droit (resp. gauche) du support compact extrémal.
In this Note, we provide a complete description of the limiting spectral measure of the Generalised Inverse Gaussian (GIG) random matrix model. The overall strategy relies on the large deviation theorem for the empirical measure of general continuous Coulomb gas. This limit is given as the extremal measure of a weighted logarithmic energy problem which may be explicitly solved here. Furthermore, we prove the almost sure convergence of the largest and smallest eigenvalues of the GIG model towards respectively the right and left-endpoints of the extremal compact support.
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@article{CRMATH_2006__342_7_519_0,
author = {F\'eral, Delphine},
title = {The limiting spectral measure of the {Generalised} {Inverse} {Gaussian} random matrix model},
journal = {Comptes Rendus. Math\'ematique},
pages = {519--522},
publisher = {Elsevier},
volume = {342},
number = {7},
year = {2006},
doi = {10.1016/j.crma.2006.01.017},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2006.01.017/}
}
TY - JOUR AU - Féral, Delphine TI - The limiting spectral measure of the Generalised Inverse Gaussian random matrix model JO - Comptes Rendus. Mathématique PY - 2006 SP - 519 EP - 522 VL - 342 IS - 7 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2006.01.017/ DO - 10.1016/j.crma.2006.01.017 LA - en ID - CRMATH_2006__342_7_519_0 ER -
%0 Journal Article %A Féral, Delphine %T The limiting spectral measure of the Generalised Inverse Gaussian random matrix model %J Comptes Rendus. Mathématique %D 2006 %P 519-522 %V 342 %N 7 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2006.01.017/ %R 10.1016/j.crma.2006.01.017 %G en %F CRMATH_2006__342_7_519_0
Féral, Delphine. The limiting spectral measure of the Generalised Inverse Gaussian random matrix model. Comptes Rendus. Mathématique, Tome 342 (2006) no. 7, pp. 519-522. doi : 10.1016/j.crma.2006.01.017. http://www.numdam.org/articles/10.1016/j.crma.2006.01.017/
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