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Universality in the Two Matrix Model with a Monomial Quartic and a General Even Polynomial Potential

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Abstract

In this paper we studied the asymptotic eigenvalue statistics of the 2 matrix model with the probability measure

$$Z^{-1}_{n}{\rm exp}\left(-n\left({\rm tr}(V(M_1)+W(M_2)-\tau M_1M_2\right)\right) \, {\rm d}M_1{\rm d} M_2,$$

in the case where \({W=\frac{y^{4}}{4}}\) and V is a general even polynomial. We studied the correlation kernel for the eigenvalues of the matrix M 1 in the limit as n → ∞. We extended the results of Duits and Kuijlaars in [14] to the case when the limiting eigenvalue density for M 1 is supported on multiple intervals. The results are achieved by constructing the parametrix to a Riemann-Hilbert problem obtained in [14] with theta functions and then showing that this parametrix is well-defined for all n by studying the theta divisor.

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Authors and Affiliations

  1. School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK

    M. Y. Mo

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Correspondence to M. Y. Mo.

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Communicated by B. Simon

The funding of this research is provided by the EPSRC grant EP/D505534/1.

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Mo, M.Y. Universality in the Two Matrix Model with a Monomial Quartic and a General Even Polynomial Potential. Commun. Math. Phys. 291, 863–894 (2009). https://doi.org/10.1007/s00220-009-0893-2

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