Completing the picture for the smallest eigenvalue of real Wishart matrices

Understand

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.