The purpose of this paper is to describe asymptotic formulas for determinants of certain operators that are analogues of Wiener-Hopf operators. The determinant formulas yield information about the distribution functions for certain random variables that arise in random matrix theory when one rescales at “the edge of the spectrum”.

We study the average case complexity of multivariate integration and L2 function approximation for the class F = C([0, 1]) of continuous functions of d variables. The class F is endowed with the isotropic Wiener measure (Brownian motion in Levy’s sense). Furthermore, for both problems, only function values are used as data.

We study asymptotic behavior of the spectrum of a Schrödinger type operator LV = L− λV on the Wiener space as λ→∞. Here L denotes the Ornstein-Uhlenbeck operator and V is a nonnegative potential function which has finitely many zero points. For some classes of potential functions, we determine the divergence order of the lowest eigenvalue. Also tunneling effect is studied.