Completing the picture for the smallest eigenvalue of real Wishart matrices

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 $W{W}^T$. 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 $\nu \ge 0$. 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.

Comments: 5 pages, 3 figuers; minor corrections; three typos corrected in comparison to the published version in PRL