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

TL;DR

This paper derives finite-size corrections to the distribution of the smallest eigenvalue in complex Wishart matrices, confirming conjectures and extending understanding of eigenvalue fluctuations at the hard and soft edges.

Contribution

It provides explicit first-order 1/N corrections to the limiting distribution of the smallest eigenvalue in Wishart matrices, using orthogonal polynomial techniques and Painlevé equations.

Findings

Explicit 1/N correction formulas for the hard edge distribution.

Confirmation of a recent conjecture by Edelman, Guionnet, and Péché.

Conjectured form of the first correction at the soft edge.

Abstract

We study the probability distribution function (PDF) of the smallest eigenvalue of Laguerre-Wishart matrices $W = X^\dagger X$ where $X$ is a random $M \times N$ ($M \geq N$) 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 $M/N \to 1$ -- 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 Painlev\'e III equation, as found by Tracy and Widom, using Fredholm operators 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 P\'ech\'e. We also study the soft edge limit, when $M-N \sim {\cal O}(N)$, for which we conjecture the form of the first correction to the limiting distribution of the smallest eigenvalue.