A solution to the string equation of unitary matrix models

Abstract

The string equations of unitary (one) matrix models (the first one is the second Painlevé equation ƒ″ −2ƒ 3 + 8zƒ=0 ) are written as consistency conditions for a system of linear equations with an irregular singularity. We argue, using methods from the theory of isomonodromic deformations, that these equations have a one parameter family of solutions which are real and finite along the real line. It is also conjectured that for a particular value of the parameter the solution behaves as z 1 2l as z → ∞. The monodromy data of the corresponding linear equations are examined and are found to be such that flows between the various models can exist.