Level spacing functions and non linear differential equations

Abstract

The Fredholm determinant of the kernel sin π(x-y)/π(x-y) on the finite interval (-t, t), appears in the theory of random matrices and in some other problems of mathematical physics. In a previous article we studied the functions S(t), A(t) and B(t) related to this Fredholm determinant, and derived relations among and differential equations for them. Here we exploit these relations to deduce the power series at t = 0, and the asymptotic behaviour at t = ∞, of the various level spacing functions of the random matrix theory.