Phase structure of matrix models through orthogonal polynomials

In the non-perturbative regime, matrix models display a large-N phase transition. For finite but large N, the transition is anticipated by strong oscillations in some coefficients in the recurrence relations for the orthogonal polynomials that allow the calculation of the partition function. The author shows how to perform the limit, requiring the definition of different interpolating functions according to the parity of polynomials, in the cases of a single or two interacting matrices.