Average characteristic polynomials in the two-matrix model

The two-matrix model is defined on pairs of Hermitian matrices [12pt]{minimal}{document}(M₁,M₂){document}(M1,M2) of size n × n by the probability measure [12pt]{minimal}{document}1/Zₙ exp (Tr (-V(M₁)-W(M₂)+τ M₁M₂)) dM₁\ dM₂,{document}1Znexp(Tr(−V(M1)−W(M2)+τM1M2))dM1dM2, where V and W are given potential functions and [12pt]{minimal}{document}τ ∈ R{document}τ∈R. We study averages of products and ratios of characteristic polynomials in the two-matrix model, where both matrices [12pt]{minimal}{document}M₁{document}M1 and [12pt]{minimal}{document}M₂{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, [12pt]{minimal}{document}pₙ(x){document}pn(x) and [12pt]{minimal}{document}qₙ(y),{document}qn(y), associated with the two-matrix model; certain transformed functions [12pt]{minimal}{document}Pₙ(w){document}P̃n(w) and [12pt]{minimal}{document}Qₙ(v){document}Q̃n(v); and finally Cauchy-type transforms of the four Eynard–Mehta kernels [12pt]{minimal}{document}K1,1{document}K1,1, [12pt]{minimal}{document}K1,2{document}K1,2, [12pt]{minimal}{document}K2,1{document}K2,1, and [12pt]{minimal}{document}K2,2{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.