Duality of Spectral Curves Arising in Two-Matrix Models

We consider the two-matrix model with the measure given by the exponential of a sum of polynomials in two different variables. We derive a sequence of pairs of dual finite-size systems of ODEs for the corresponding biorthonormal polynomials. We prove an inverse theorem, which shows how to reconstruct such measures from pairs of semi-infinite finite-band matrices, which define the recursion relations and satisfy the string equation. In the limit $N\to\infty$, we prove that the obtained dual systems have the same spectral curve.