Abstract
Rectangular real matrices 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 . The extreme eigenvalues of are of particular interest. We explicitly compute the distribution and the gap probability of the smallest nonzero eigenvalue in this ensemble, both for arbitrary fixed and , and in the universal large limit with fixed. We uncover an integrable Pfaffian structure valid for all even values of . This extends previous results for odd at infinite and recursive results for finite and for all . Our mathematical results include the computation of expectation values of half-integer powers of characteristic polynomials.
- Received 1 September 2014
DOI:https://doi.org/10.1103/PhysRevLett.113.250201
© 2014 American Physical Society