The Hermitian two matrix model with an even quartic potential (Q2882489)

The Hermitian two matrix model is represented by a probability measure of the form NEWLINE\[NEWLINE\frac{1}{Z_n}(-n \text{Tr}(V(M_1) + W(M_2) - \tau M_1 M_2))dM_1 dM_2,NEWLINE\]NEWLINE defined on the space of pairs \((M_1, M_2)\) of \(n \times n\) Hermitian matrices. In this formula, \(Z_n\) is a normalization constant, \(\tau \in {\mathbb R}\backslash \{0\}\) is a coupling constant, \(dM_1 dM_2\) is the flat Lebesgue measure on the space of pairs of Hermitian matrices and \(V\), \(W\) are the potentials of the matrix model.NEWLINENEWLINEIn this paper, the potential \(V\) is considered to be even polynomial and \(W\) has the form \(W(y)=y^{4}/4 + \alpha y^{2}/2\), \(\alpha \in {\mathbb R}\). Then, the description of the eigenvalues of \(M_1\) in the large \(n\) limit is studied. For that purpose a vector equilibrium problem characterizing the limiting mean density for the eigenvalues of \(M_1\) is formulated. The eigenvalues of the matrices \(M_1\) and \(M_2\) in the two matrix model are a determinantal point process with correlation kernels that are expressed in terms of biorthogonal polynomials. The formulation of a Riemann-Hilbert problem here for biorthogonal polynomials offers new possibilities for the steepest descent method which was applied already very successfully to the Riemann-Hilbert problem for orthogonal polynomials in series of works. The application of the steepest descent method to the Riemann-Hilbert problem of \(4 \times 4\) matrices gives a precise asymptotic analysis of these kernels. The presented results are the generalization of results of \textit{M. Duits} and \textit{A. B. J. Kuijlaars} [Commun. Pure Appl. Math. 62, No. 8, 1076--1153 (2009; Zbl 1221.15052)] with \(\alpha = 0\) to the case of general \(\alpha\).