Abstract
We discuss, perturbatively and nonperturbatively, the multiband phase structure that arises in Hermitian one-matrix models with potentials having several local minima. The tree-level phase diagram for the potential including critical exponents at the phase boundaries is presented. The multiband structure is then studied from the viewpoint of the orthogonal polynomial recursion coefficients , using the operator formalism to relate them to the large- limit of the generating function . We show how a periodicity structure in the sequence of the coefficients naturally leads to multiband structure, and in particular, provide an explicit example of a three-band phase. Numerical evidence for the periodicity structure among the recursion coefficients is given. We then present examples where we identify the double-scaling limit from a multiband phase. In particular, a -type multicritical nonperturbative solution from the two-band phase in the potential, and a -type nonperturbative solution from the three-band phase in the potential is found. Both solutions are described by differential equations related to the modified Korteweg-de Vries hierarchy. Finally, we comment on the other phases that coexist with the pure gravity solution.
- Received 17 August 1990
DOI:https://doi.org/10.1103/PhysRevD.42.4105
©1990 American Physical Society