Asymptotics of the partition function of a random matrix model

Nous prouvons de nombreux resultats concernant les comportements asymptotiques de l'energie libre d'un modele matriciel aleatoire a potentiel polynomial. Notre approche est fondee sur la deformation du potentiel et de l'utilisation de la structure integrable sous-jacente du modele. Les principaux resultats incluent l'existence du developpement asymptotique en puissances de N impaires des coefficients de recurrence des polynomes orthogonaux d'un potentiel regulier a une coupe et de la double reduction asymptotique de l'energie libre pour un potentiel quartique singulier. Nous prouvons aussi l'analyticite des coefficients du developpement asymptotique des coefficients de recurrence et de l'energie selon ceux du potentiel libre, ainsi que l'analyticite unilaterale des coefficients et de l'energie libre d'un potentiel singulier a une coupe.

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