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

Abstract

We study the probability density function (PDF) of the smallest eigenvalue of Laguerre–Wishart matrices [Formula: see text] where [Formula: see text] is a random [Formula: see text] ([Formula: see text]) 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 [Formula: see text], large [Formula: see text] with [Formula: see text] — i.e. for quasi-square large matrices [Formula: see text] — we show that this PDF, in the hard edge limit, can be expressed in terms of the solution of a Painlevé III equation, as found by Tracy and Widom, using Fredholm operator techniques. Furthermore, our method allows us to compute explicitly the first [Formula: see text] corrections to this limiting distribution at the hard edge. Our computations confirm a recent conjecture by Edelman, Guionnet and Péché. We also study the soft edge limit, when [Formula: see text], for which we conjecture the form of the first correction to the limiting distribution of the smallest eigenvalue.