Unitary-matrix models as exactly solvable string theories

Models of unitary matrices are solved exactly in a double scaling limit, using orthogonal polynomials on a circle. Exact differential equations are found for the scaling functions of these models. For the simplest model (k=1), the Painlevé II equation with constant 0 is obtained. There are possible nonperturbative phase transitions in these models. The scaling function is of the form ${\mathit{N}}^{\mathrm{-}1/(2\mathit{k}+1)}$×f(${\mathit{N}}^{2\mathit{k}/(2\mathrm{}\mathit{k}+1)}$(${\lambda }_{\mathit{x}}$-λ)) for the kth multicritical point. The specific heat is ${\mathit{f}}^2$, and is therefore manifestly positive. Equations are given for k=2 and 3, with a discussion of asymptotic behavior.