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 exp n P m k=1 s jWe 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 + vnp n−1 ) = p n+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/u n−1 = −x n−1 /xn satisfies the PainlevéIII equation; and vn/(vn − un) = 1 − x −2 n 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.

Eigenvalue density in Hermitian matrix models by the Lax pair method

“…This paper is a continuation of the previous works [1][2][3][4] about the linearized equation d 2 φ/dη 2 = −ξ 2 F (η, ξ)φ for the Painlevé or discrete Painlevé equations. The connection between the integral η η 0 √ F (t, ξ) dt in the WKB asymptotics and the analytic potential in the previous researches is now extended to the relation between √ F (η, ξ) and the derivative of the potential function in the complex plane to investigate the distribution of eigenvalues considered in matrix models.…”

“…As a remark, if W (η) = g 3 η 3 + g 4 η 4 is degenerated to W (η) = g 4 η 4 by taking a → 0, (7.9) becomes E (0) = 3/8 − ln b. We will see next that E (0) has the same result as W (η) = g 2 η 2 + g 4 η 4 is degenerated to W (η) = g 4 η 4 . We can also use (7.1) to get other special densities for m = 2.…”

“…Consider the orthogonal polynomials p n (z) = z n + • • • on the unit circle |z| = 1 with the potential function V (z) = s(z + z −1 ) [4,37]:…”

“…], which is the well-known Wigner semicircle. When m = 2 and W (η) = 1 2 η 2 + gη 4 , it is given in [6] that…”

“…Other models can also be studied by the Lax pair method as seen in the appendix. The weak-and strong-coupling densities in the unitary matrix model [20] can be derived using the Lax pair for the discrete Painlevé II equation [4] associated with the orthogonal polynomials on unit circle. The density in [22,23] can be obtained using the Laguerre polynomials.…”