FREDHOLM DETERMINANTS, DIFFERENTIAL-EQUATIONS AND MATRIX MODELS

Orthogonal polynomial random matrix models of N x N hermitian matrices lead to Fredholm determinants of integral operators with kernel of th e form (phi(x)psi(y)-psi(x)phi(y))/x-y. This paper is concerned with t he Fredholm determinants of integral operators having kernel of this f orm and where the underlying set is the union of intervals J = or(j=1) m (a2j-1, a2j). The emphasis is on the determinants thought of as func tions of the end-points a(k). We show that these Fredholm determinants with kernels of the general form described above are expressible in t erms of solutions of systems of PDE's as long as phi and psi satisfy a certain type of differentiation formula. The (phi, psi) pairs for the sine, Airy, and Bessel kernels satisfy such relations, as do the pair s which arise in the finite N Hermite, Laguerre and Jacobi ensembles a nd in matrix models of 2D quantum gravity. Therefore we shall be able to write down the systems of PDE's for these ensembles as special case s of the general system. An analysis of these equations will lead to e xplicit representations in terms of Painleve transcendents for the dis tribution functions of the largest and smallest eigenvalues in the fin ite N Hermite and Laguerre ensembles, and for the distribution functio ns of the largest and smallest singular values of rectangular matrices (of arbitrary dimensions) whose entries are independent identically d istributed complex Gaussian variables. There is also an exponential va riant of the kernel in which the denominator is replaced by e(bx)-e(by ), where b is an arbitrary complex number. We shall find an analogous system of differential equations in this setting. If b = i then we can interpret our operator as acting on (a subset of) the unit circle in the complex plane. As an application of this we shall write down a sys tem of PDE's for Dyson's circular ensemble of N x N unitary matrices, and then an ODE if J is an arc of the circle.