Average characteristic polynomials in the two-matrix model

Abstract

The two-matrix model is defined on pairs of Hermitian matrices \documentclass[12pt]{minimal}\begin{document}$(M_1,M_2)$\end{document}(M1,M2) of size n × n by the probability measure \documentclass[12pt]{minimal}\begin{document}$\frac{1}{Z_n} \exp (\hbox{Tr} (-V(M_1)-W(M_2)+\tau M_1M_2))$\break $ dM_1\ dM_2,$\end{document}1Znexp(Tr(−V(M1)−W(M2)+τM1M2))dM1dM2, where V and W are given potential functions and \documentclass[12pt]{minimal}\begin{document}$\tau \in \mathbb {R}$\end{document}τ∈R. We study averages of products and ratios of characteristic polynomials in the two-matrix model, where both matrices \documentclass[12pt]{minimal}\begin{document}$M_1$\end{document}M1 and \documentclass[12pt]{minimal}\begin{document}$M_2$\end{document}M2 may appear in a combined way in both numerator and denominator. We obtain determinantal expressions for such averages. The determinants are constructed from several building blocks: the biorthogonal polynomials, \documentclass[12pt]{minimal}\begin{document}$p_n(x)$\end{document}pn(x) and \documentclass[12pt]{minimal}\begin{document}$q_n(y),$\end{document}qn(y), associated with the two-matrix model; certain transformed functions \documentclass[12pt]{minimal}\begin{document}$\widetilde{P}_n(w)$\end{document}P̃n(w) and \documentclass[12pt]{minimal}\begin{document}$\widetilde{Q}_n(v)$\end{document}Q̃n(v); and finally Cauchy-type transforms of the four Eynard–Mehta kernels \documentclass[12pt]{minimal}\begin{document}$K_{1,1}$\end{document}K1,1, \documentclass[12pt]{minimal}\begin{document}$K_{1,2}$\end{document}K1,2, \documentclass[12pt]{minimal}\begin{document}$K_{2,1}$\end{document}K2,1, and \documentclass[12pt]{minimal}\begin{document}$K_{2,2}$\end{document}K2,2. In this way, we generalize known results for the one-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.