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Character expansion method for the first order asymptotics of a matrix integral
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  • Published: 10 February 2005

Character expansion method for the first order asymptotics of a matrix integral

  • Alice Guionnet1 &
  • Mylène Maïda2 

Probability Theory and Related Fields volume 132, pages 539–578 (2005)Cite this article

Abstract.

The estimation of various matrix integrals as the size of the matrices goes to infinity is motivated by theoretical physics, geometry and free probability questions. On a rigorous ground, only integrals of one matrix or of several matrices with simple quadratic interaction (called AB interaction) could be evaluated so far (see e.g. [19], [17] or [9]). In this article, we follow an idea widely developed in the physics literature, which is based on character expansion, to study more complex interaction. In this context, we derive a large deviation principle for the empirical measure of Young tableaux. We then use it to study a matrix model defined in the spirit of the ‘dually weighted graph model’ introduced in [13], but with a cutoff function such that the matrix integral and its character expansion converge. We prove that the free energy of this model converges as the size of the matrices goes to infinity and study the critical points of the limit.

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Authors and Affiliations

  1. Ecole Normale Supérieure de Lyon, Unité de Mathématiques pures et appliquées, UMR 5669, 46 Allée d’Italie, 69364, Lyon Cedex 07, France

    Alice Guionnet

  2. Laboratoire Modal-X, Université Paris X-Nanterre, 200 av. de la république, 92001, Nanterre Cedex, France

    Mylène Maïda

Authors
  1. Alice Guionnet
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  2. Mylène Maïda
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Correspondence to Mylène Maïda.

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Guionnet, A., Maïda, M. Character expansion method for the first order asymptotics of a matrix integral. Probab. Theory Relat. Fields 132, 539–578 (2005). https://doi.org/10.1007/s00440-004-0403-6

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  • Received: 18 January 2004

  • Revised: 30 September 2004

  • Published: 10 February 2005

  • Issue Date: July 2005

  • DOI: https://doi.org/10.1007/s00440-004-0403-6

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Keywords

  • Large deviations
  • Random matrices
  • Non-commutative measure
  • Integration
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