Matrix kernels for the Gaussian orthogonal and symplectic ensembles

where A(x) = Ai(x) is the Airy function. (See [5], [11] for recent reviews.) Typically these results are proved by first establishing that for finite N the distribution of the right-most particle is the Fredholm determinant of an operator KN . The limit theorem then follows once one proves KN → KAiry in trace norm. The classic example is the finite N Gaussian Unitary Ensemble (GUE) where KN is the Hermite kernel [6] and the limit N → ∞ is the edge scaling limit of the largest eigenvalue. In this case the unitary invariance of the underlying probability measure is manifest.