Periodic regularization, multiband structure, and orthogonal polynomials

Instead of using polynomial potentials for Hermitian matrix models which are bounded from below, an alternative regularization scheme is to make the potential periodic by employing unitary matrices. We discuss, perturbatively and nonperturbatively, the multiband phase structure that arises in unitary one-matrix models with potentials having several local minima. The tree-level phase diagram for a prototypic U2(θ) potential is presented. The multiband structure is then studied from the viewpoint of the orthogonal polynomial recursion coefficients Tn, using the operator formalism to relate them to the large-N limit of the generating function F(z) ==-(i/N)left-angle Tr(1+zU)/(1-zU)right-angle. We show how a periodicity structure in the sequence of the Tn coefficients naturally leads to multiband structure. We next study the double-scaling limit from various phase boundaries leading to nonperturbative string equations for unitary matrix models. Finally, we comment on the fate of the k=2 pure gravity solution under a periodic regularization