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Universality in several-matrix models via approximate transport maps

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Acta Mathematica

Abstract

We construct approximate transport maps for perturbative several-matrix models. As a consequence, we deduce that local statistics have the same asymptotic as in the case of independent GUE or GOE matrices (i.e., they are given by the sine-kernel in the bulk and the Tracy–Widom distribution at the edge), and we show averaged energy universality (i.e., universality for averages of m-points correlation functions around some energy level E in the bulk). As a corollary, these results yield universality for self-adjoint polynomials in several independent GUE or GOE matrices which are close to the identity.

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Authors and Affiliations

  1. Department of Mathematics, ETH Zürich, Rämistrasse 101, CH-8092, Zürich, Switzerland

    Alessio Figalli

  2. Université de Lyon, École Normale Supérieure de Lyon, site Monod, UMPA UMR 5669 CNRS, 46, allée d’Italie, FR-69364, Lyon Cedex 07, France

    Alice Guionnet

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  1. Alessio Figalli
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  2. Alice Guionnet
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Figalli, A., Guionnet, A. Universality in several-matrix models via approximate transport maps. Acta Math 217, 81–176 (2016). https://doi.org/10.1007/s11511-016-0142-4

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  • DOI: https://doi.org/10.1007/s11511-016-0142-4

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