Journal of Mathematical Physics vol:52 issue:1 pages:-
The two-matrix model is defined on pairs of Hermitian matrices (M1, M2) of size n x n by the probability measure 1/Z(n) exp(Tr(-V(M1)-W(M2) + tau M1M2)) dM1 dM2, where V and W are given potential functions and tau is an element of R. We study averages of products and ratios of characteristic polynomials in the two-matrix model, where both matrices M1 and 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, pn(x) and qn(y), associated with the two-matrix model; certain transformed functions Pn(w) and Qn(v); and finally Cauchy-type transforms of the four Eynard-Mehta kernels K1,K1, K1,K2, K2,K1, and K2,K2. 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.