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

Understand

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.