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1. Two-matrix models

This section contains open problems and questions related to the two-matrix models

    1. Two matrix model with non symmetric potentials


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      Problem 1.1.

      [M. Bertola (after M. Duits)] Study eigenvalue properties (in particular, the limiting density of eigenvalues) of $M_1$ in a two matrix model given by the probability density $$ \frac{1}{Z_N}e^{-N\textrm{tr}[V(M_1)+W(M_2)+tM_1M_2]}dM_1 dM_2. $$ In this context, $V(x)=x^2/2$ and $W(y)$ is a general polynomial of even degree (no symmetry). More in general, consider this model when $V(x)$ is an even quartic polynomial and $W(y)$ is a general polynomial with no symmetries.
          A more general question would be the following: if it is possible to obtain information about the eigenvalues of $M_1$ and $M_2$ in a two-matrix model, and the analysis is significantly simpler for one of the two matrices, could this be used for studying the eigenvalues of the other one?

        • Coupled random matrix model with negative potentials


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          Problem 1.2.

          [M. Duits] Analysis of the coupled matrix model similar to problem two-matrix-x2, but with potentials $$ V(x)=-(x^4-ax^2), \qquad W(y)=V(y). $$ Is there any new critical behavior for some value of $a$?

            • $2+1/2$ random matrix models


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              Problem 1.3.

              [A. Martínez-Finkelshtein] A general question about Riemann-Hilbert techniques to study the two-matrix plus external source model.

                  Cite this as: AimPL: Vector equilibrium problems and random matrix models, available at http://aimpl.org/vectorequilib.