Average characteristic polynomials in the two-matrix model

Abstract

The two-matrix model is defined on pairs of Hermitian matrices $(M_1,M_2)$ of size $n\times n$ by the probability measure $$\frac{1}{Z_n}\exp \left(\text{Tr}(-V(M_1)-W(M_2)+\tau M_1{M}_2)\right)d{M}_{1}d{M}_2,$$ where $V$ and $W$ are given potential functions and $τ\in\er$. We study averages of products and ratios of characteristic polynomials in the two-matrix model, where both matrices $M_1$ and $M_2$ may appear in a combined way in both numerator and denominator. We obtain determinantal expressions for such averages. The determinants are constructed from several building blocks: the biorthogonal polynomials $p_n(x)$ and $q_n(y)$ associated to the two-matrix model; certain transformed functions ${\P }_n(w)$ and $\Q_n(v)$; and finally Cauchy-type transforms of the four Eynard-Mehta kernels $K_{1,1}$, $K_{1,2}$, $K_{2,1}$ and $K_{2,2}$. In this way we generalize known results for the $1$-matrix model. Our results also imply a new proof of the Eynard-Mehta theorem for correlation functions in the two-matrix model, and they lead to a generating function for averages of products of traces.

28 pages, references added

References in corpus (6)

• Differential Equations for Dyson Processes
• The Cauchy two-matrix model
• Derivation of determinantal structures for random matrix ensembles in a new way
• A new Chiral Two-Matrix Theory for Dirac Spectra with Imaginary Chemical Potential
• Biorthogonal Polynomials for Potentials of two Variables and External Sources at the Denominator
• Mixed correlation function and spectral curve for the 2-matrix model

Cited by in corpus (3)

• Bessel process, Schramm-Loewner evolution, and Dyson model
• The Hermitian two matrix model with an even quartic potential
• Characteristic Polynomials of Random Matrices and Noncolliding Diffusion Processes