Completing the picture for the smallest eigenvalue of real Wishart matrices

Rectangular real $N times (N + u)$ 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 $ u$, and in the universal large $N$ limit with $ u$ fixed. We uncover an integrable Pfaffian structure valid for all even values of $ ugeq 0$. This extends previous results for odd $ u$ at infinite $N$ and recursive results for finite $N$ and for all $ u$. Our mathematical results include the computation of expectation values of half integer powers of characteristic polynomials.