The Widom–Dyson constant for the gap probability in random matrix theory

In the bulk scaling limit for the Gaussian Unitary Ensemble in random matrix theory, the probability that there are no eigenvalues in the interval (0,2s)(0,2s) is given by Ps=det(I-Ks)Ps=det(I-Ks), where KsKs is the trace-class operator with kernel Ks(x,y)=sin(x-y)π(x-y) acting on L2(0,2s)L2(0,2s). In the analysis of the asymptotic behavior of PsPs as s→∞s→∞, there is particular interest in the constant term known as the Widom–Dyson constant. We present a new derivation of this constant, which can be adapted to calculate similar critical constants in other problems arising in random matrix theory.