Phys. Rev. Lett. 64, 1326–1329 (1990)Unitary-matrix models as exactly solvable string theoriesReceived 20 December 1989; published in the issue dated 19 March 1990 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 N-1/(2k+1)×f(N2k/(2k+1)(λx-λ)) for the kth multicritical point. The specific heat is f2, and is therefore manifestly positive. Equations are given for k=2 and 3, with a discussion of asymptotic behavior. © 1990 The American Physical Society URL:
http://link.aps.org/doi/10.1103/PhysRevLett.64.1326
DOI:
10.1103/PhysRevLett.64.1326
PACS:
11.17.+y, 05.90.+m, 11.15.Pg, 11.15.Tk
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