DISCRETE INTEGRABLE SYSTEMS ASSOCIATED WITH THE UNITARY MATRIX MODEL
Abstract
The orthogonal polynomials on the unit circle associated with the unitary matrix model have various interesting properties, and have been studied in connection with different applications, such as double scaling, Riemann–Hilbert problems and integrable Fredholm operators. In this paper, we study the orthogonal polynomials on the unit circle with the weight function 
. We use the orthogonality of the polynomials to show that the orthogonal polynomials on the unit circle satisfy the linear problems associated with the discrete PainlevéII hierarchy, alternate discrete PainlevéII hierarchy and the discrete MKdV hierarchy. Thus the isomonodromy deformation method can be used to study the unitary matrix model. Typically, we focus on the orthogonal polynomials with the weight function exp{s(z+z-1)}. The recursion formula z(pn+vnpn-1)=pn+1+unpn (derived from Szegö's equation) for the orthogonal polynomials pn(z,s) is proved to be equivalent to the discrete AKNS-ZS system which is the base equation in the linear problem for the discrete equations. The variable xn=pn(0,s) satisfies the discrete PainlevéII equation and the discrete MKdV equation; un, vn satisfy the alternate discrete PainlevéII equation; 1/un-1=-xn-1/xn satisfies the PainlevéIII equation; and 
satisfies the PainlevéV equation. Also, we discuss the continuum limits of the discrete equations by using the double scaling method. Further, we present a procedure for finding the linear equations for the orthogonal polynomials and the consistency conditions of the linear equations by using the orthogonality of the polynomials.
