Completing the picture for the smallest eigenvalue of real Wishart matrices

TL;DR

This paper derives explicit formulas for the distribution and gap probability of the smallest non-zero eigenvalue of real Wishart matrices, revealing an integrable Pfaffian structure for all even  values of .

Contribution

It extends previous results by providing explicit distributions and gap probabilities for the smallest eigenvalue of real Wishart matrices for all finite sizes and even  values of , including a new Pfaffian structure.

Findings

Derived explicit distribution and gap probability formulas.

Uncovered an integrable Pfaffian structure for even  .

Extended previous results to all finite sizes and even   values.

Abstract

Rectangular real $N \times (N + \nu)$ matrices $W$ with a Gaussian distribution appear very frequently in data analysis, condensed matter physics and quantum field theory. A central question concerns the correlations encoded in the spectral statistics of $WW^T$. The extreme eigenvalues of $W W^T$ are of particular interest. We explicitly compute the distribution and the gap probability of the smallest non-zero eigenvalue in this ensemble, both for arbitrary fixed $N$ and $\nu$, and in the universal large $N$ limit with $\nu$ fixed. We uncover an integrable Pfaffian structure valid for all even values of $\nu\geq 0$. This extends previous results for odd $\nu$ at infinite $N$ and recursive results for finite $N$ and for all $\nu$. Our mathematical results include the computation of expectation values of half integer powers of characteristic polynomials.