The Smallest Eigenvalue Distribution in the Real Wishart-Laguerre Ensemble with Even Topology

Understand

We consider rectangular random matrices of size $p\times n$ belonging to the real Wishart-Laguerre ensemble also known as the chiral Gaussian orthogonal ensemble.

• This ensemble appears in many applications like QCD, mesoscopic physics, and time series analysis.
• We are particularly interested in the distribution of the smallest non-zero eigenvalue and the gap probability to find no eigenvalue in an interval $[0,t]$.
• While for odd topology $\nu=n-p$ explicit closed results are known for finite and infinite matrix size, for even $\nu>2$ only recursive expressions in $p$ are available.The smallest eigenvalue distribution as well as the gap probability for general even $\nu$ is equivalent to expectation values of characteristic polynomials raised to a half-integer.