PUB - Publikationen an der Universität Bielefeld

Rectangular real $N \times (N + \nu)$ matrices $W$ with a Gaussiandistribution appear very frequently in data analysis, condensed matter physicsand quantum field theory. A central question concerns the correlations encodedin the spectral statistics of $WW^T$. The extreme eigenvalues of $W W^T$ are ofparticular interest. We explicitly compute the distribution and the gapprobability of the smallest non-zero eigenvalue in this ensemble, both forarbitrary 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 $N$ andrecursive results for finite $N$ and for all $\nu$. Our mathematical resultsinclude the computation of expectation values of half integer powers ofcharacteristic polynomials.