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

TL;DR

This paper derives explicit formulas for the smallest eigenvalue distribution and gap probability in the real Wishart-Laguerre ensemble with even topology, using advanced mathematical techniques, and confirms universality through numerical simulations.

Contribution

It provides the first explicit Pfaffian formulas for even topology cases in the real Wishart-Laguerre ensemble, extending known results from odd topology.

Findings

Explicit Pfaffian formulas for even topology eigenvalue distributions.

Universality of results confirmed through numerical simulations.

Results applicable in physics, statistics, and related fields.

Abstract

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. The computation of such averages is done via a combination of skew-orthogonal polynomials and bosonisation methods. The results are given in terms of Pfaffian determinants both at finite $p$ and in the hard edge scaling limit ($p\to\infty$ and $\nu$ fixed) for an arbitrary even topology $\nu$. Numerical simulations for the correlated Wishart ensemble illustrate the universality of our results in this particular limit. These simulations point to a validity of the hard edge scaling limit beyond the invariant case.