  We examine the phase structure and the critical processes of the spectral
curves that arise in the study of large N dualities between supersymmetric
Yang-Mills theories and string models on local Calabi-Yau manifolds. These
spectral curves are determined by a set of complex partial 't Hooft parameters
and a system of cuts given by projections on the spectral curve of minimal
supersymmetric cycles of the underlying Calabi-Yau manifold. Using a
combination of analytical and numerical methods we give a complete description
of the one-cut phase in the cubic model, determine the analytic condition
satisfied by critical one-cut spectral curves, and give an algorithm to
calculate the two-cut spectral curves of the cubic model for generic values of
the partial 't Hooft parameters.
