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Asymptotic Expansion of β Matrix Models in the One-cut Regime

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Abstract

We prove the existence of a 1/N expansion to all orders in β matrix models with a confining, offcritical potential corresponding to an equilibrium measure with a connected support. Thus, the coefficients of the expansion can be obtained recursively by the “topological recursion” derived in Chekhov and Eynard (JHEP 0612:026, 2006). Our method relies on the combination of a priori bounds on the correlators and the study of Schwinger-Dyson equations, thanks to the uses of classical complex analysis techniques. These a priori bounds can be derived following (Boutet de Monvel et al. in J Stat Phys 79(3–4):585–611, 1995; Johansson in Duke Math J 91(1):151–204, 1998; Kriecherbauer and Shcherbina in Fluctuations of eigenvalues of matrix models and their applications, 2010) or for strictly convex potentials by using concentration of measure (Anderson et al. in An introduction to random matrices, Sect. 2.3, Cambridge University Press, Cambridge, 2010). Doing so, we extend the strategy of Guionnet and Maurel-Segala (Ann Probab 35:2160–2212, 2007), from the hermitian models (β = 2) and perturbative potentials, to general β models. The existence of the first correction in 1/N was considered in Johansson (1998) and more recently in Kriecherbauer and Shcherbina (2010). Here, by taking similar hypotheses, we extend the result to all orders in 1/N.

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Authors and Affiliations

  1. Institut de Physique Théorique de Saclay, 91191, Gif-sur-Yvette Cedex, France

    G. Borot

  2. UMPA, CNRS UMR 5669, ENS Lyon, 46 allée d’Italie, 69007, Lyon, France

    A. Guionnet

  3. Section de Mathematiques, Universite de Geneve, 2-4 rue du Lievre, 1211, Geneve, Switzerland

    G. Borot

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  1. G. Borot
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  2. A. Guionnet
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Correspondence to A. Guionnet.

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Communicated by H.-T. Yau

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Borot, G., Guionnet, A. Asymptotic Expansion of β Matrix Models in the One-cut Regime. Commun. Math. Phys. 317, 447–483 (2013). https://doi.org/10.1007/s00220-012-1619-4

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  • DOI: https://doi.org/10.1007/s00220-012-1619-4

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