Multiband structure and critical behavior of matrix models

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 ${\phi }^6$ 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 $R_r$, using the operator formalism to relate them to the large-$N$ limit of the generating function $F\left(z\right)\mathrm{}\equiv \left(\frac{1}{N}\right)〈\frac{\mathrm{tr}1}{(z-\Phi )}〉$. We show how a periodicity structure in the sequence of the $R_n$ 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 $k=2\mathrm{}$-type multicritical nonperturbative solution from the two-band phase in the ${\phi }^8$ potential, and a $k=1\mathrm{}$-type nonperturbative solution from the three-band phase in the ${\phi }^6$ 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 $k=2\mathrm{}$ pure gravity solution.