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Strong asymptotics for Cauchy biorthogonal polynomials with application to the Cauchy two-matrix model

We apply the nonlinear steepest descent method to a class of 3 × 3 Riemann-Hilbert problems introduced in connection with the Cauchy two-matrix random model. The general case of two equilibrium measures supported on an arbitrary number of intervals is considered. In this case, we solve the Riemann-Hilbert problem for the outer parametrix in terms of sections of a spinorial line bundle on a three-sheeted Riemann surface of arbitrary genus and establish strong asymptotic results for the Cauchy biorthogonal polynomials.