The smallest eigenvalue distribution in the real WishartLaguerre ensemble with even topology
Abstract
We consider rectangular random matrices of size p× n belonging to the real WishartLaguerre 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 nonzero eigenvalue and the gap probability to find no eigenvalue in an interval [0,t]. While for odd topology ν =np explicit closed results are known for finite and infinite matrix size, for even ν \gt 2 only recursive expressions in p are available. The smallest eigenvalue distribution as well as the gap probability for general even ν is equivalent to expectation values of characteristic polynomials raised to a halfinteger power. The computation of such averages is done via a combination of skeworthogonal 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 ∞ and ν fixed) for an arbitrary even topology ν. 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.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 June 2015
 DOI:
 10.1088/17518113/48/24/245202
 arXiv:
 arXiv:1502.03685
 Bibcode:
 2015JPhA...48x5202W
 Keywords:

 Mathematical Physics;
 Condensed Matter  Statistical Mechanics;
 High Energy Physics  Lattice
 EPrint:
 28 pages, 5 figures