Finite N corrections to the limiting distribution of the smallest eigenvalue of Wishart complex matrices

We study the probability density function (PDF) of the smallest eigenvalue of Laguerre-Wishart matrices W = (XX)-X-dagger where X is a random M x N (M >= N) matrix, with complex Gaussian independent entries. We compute this PDF in terms of semi-classical orthogonal polynomials, which are deformations of Laguerre polynomials. By analyzing these polynomials, and their associated recurrence relations, in the limit of large N, large M with M/N -> 1 - i.e. for quasi-square large matrices X - we show that this PDF, in the hard edge limit, can be expressed in terms of the solution of a Painleve III equation, as found by Tracy and Widom, using Fredholm operator techniques. Furthermore, our method allows us to compute explicitly the first 1/N corrections to this limiting distribution at the hard edge. Our computations confirm a recent conjecture by Edelman, Guionnet and Peche. We also study the soft edge limit, when M - N similar to O(N), for which we conjecture the form of the first correction to the limiting distribution of the smallest eigenvalue.