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The Cauchy Two-Matrix Model

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Abstract

We introduce a new class of two(multi)-matrix models of positive Hermitian matrices coupled in a chain; the coupling is related to the Cauchy kernel and differs from the exponential coupling more commonly used in similar models. The correlation functions are expressed entirely in terms of certain biorthogonal polynomials and solutions of appropriate Riemann–Hilbert problems, thus paving the way to a steepest descent analysis and universality results. The interpretation of the formal expansion of the partition function in terms of multicolored ribbon-graphs is provided and a connection to the O(1) model. A steepest descent analysis of the partition function reveals that the model is related to a trigonal curve (three-sheeted covering of the plane) much in the same way as the Hermitian matrix model is related to a hyperelliptic curve.

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Authors and Affiliations

  1. Centre de recherches mathématiques, Université de Montréal, C.P. 6128, succ. centre ville, Montréal, Québec, Canada, H3C 3J7

    M. Bertola

  2. Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve W., Montréal, Québec, Canada, H3G 1M8

    M. Bertola

  3. Department of Mathematics, 255 Hurley Hall, Notre Dame, IN, 46556-4618, USA

    M. Gekhtman

  4. Department of Mathematics and Statistics, University of Saskatchewan, 106 Wiggins Road, Saskatoon, Saskatchewan, S7N 5E6, Canada

    J. Szmigielski

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  1. M. Bertola
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  3. J. Szmigielski
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Correspondence to M. Bertola.

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Communicated by L. Takhtajan

Work supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC).

Work supported in part by NSF Grant DMD-0400484.

Work supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC), Grant. No. 138591-04.

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Bertola, M., Gekhtman, M. & Szmigielski, J. The Cauchy Two-Matrix Model. Commun. Math. Phys. 287, 983–1014 (2009). https://doi.org/10.1007/s00220-009-0739-y

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