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

Abstract

We study the probability distribution function (PDF) of the smallest eigenvalue of Laguerre-Wishart matrices $W=X^{†}X$ where $X$ is a random $M\times N$ ($M\ge N$) 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 Painlevé 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 Péché. We also study the soft edge limit, when $M-N\sim \mathcal{O}(N)$, for which we conjecture the form of the first correction to the limiting distribution of the smallest eigenvalue.

28 pages, 2 Figures

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• Optimal soft edge scaling variables for the Gaussian and Laguerre even $\beta$ ensembles
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• A Survey on the Eigenvalues Local Behavior of Large Complex Correlated Wishart Matrices
• Finite size corrections for real eigenvalues of the elliptic Ginibre matrices
• Extreme Eigenvalue Distributions of Jacobi Ensembles: New Exact Representations, Asymptotics and Finite Size Corrections
• Volume of the set of LOCC-convertible quantum states
• Hole probability for noninteracting fermions in a $d$-dimensional trap
• Asymptotic expansions relating to the distribution of the length of longest increasing subsequences
• Asymptotics of the Largest Eigenvalue Distribution of the Laguerre Unitary Ensemble
• Rate of convergence at the hard edge for various Pólya ensembles of positive definite matrices
• Tracy-Widom distributions for the Gaussian orthogonal and symplectic ensembles revisited: a skew-orthogonal polynomials approach