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, 2 s) is given by Ps = det (I - Ks), where Ks is the trace-class operator with kernel Ks (x, y) = frac(sin (x - y), π (x - y)) acting on L2 (0, 2 s). In the analysis of the asymptotic behavior of Ps as 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. © 2006 Elsevier B.V. All rights reserved.