Asymptotics of the partition function of a random matrix model

M. Bleher, Pavel; Its, Alexander

Annales de l'Institut Fourier, Tome 55 (2005) no. 6, pp. 1943-2000.

Nous prouvons de nombreux résultats concernant les comportements asymptotiques de l’énergie libre d’un modèle matriciel aléatoire à potentiel polynômial. Notre approche est fondée sur la déformation du potentiel et de l’utilisation de la structure intégrable sous-jacente du modèle. Les principaux résultats incluent l’existence du développement asymptotique en puissances de $N$ impaires des coefficients de récurrence des polynômes orthogonaux d’un potentiel régulier à une coupe et de la double réduction asymptotique de l’énergie libre pour un potentiel quartique singulier. Nous prouvons aussi l’analyticité des coefficients du développement asymptotique des coefficients de récurrence et de l’énergie selon ceux du potentiel libre, ainsi que l’analyticité unilatérale des coefficients et de l’énergie libre d’un potentiel singulier à une coupe.

We prove a number of results concerning the large $N$ asymptotics of the free energy of a random matrix model with a polynomial potential. Our approach is based on a deformation of potential and on the use of the underlying integrable structures of the matrix model. The main results include the existence of a full asymptotic expansion in even powers of $N$ of the recurrence coefficients of the related orthogonal polynomials for a one-cut regular potential and the double scaling asymptotics of the free energy for a singular quartic potential. We also prove the analyticity of the coefficients of the asymptotic expansions of the recurrence coefficients and the free energy, with respect to the coefficients of the potential, and the one-sided analyticity of the recurrent coefficients and the free energy for a one-cut singular potential.

Classification : 42C05
Keywords: Matrix Models, orthogonal polynomials, partition function
Mot clés : modèles matriciels, polynômes orthogonaux, fonction de partition

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