A few results and conjectures on some product-ratio correlation functions of characteristic polynomials of beta-Hermite ensembles

Rank-one non-Hermitian deformations of  tridiagonal beta-Hermite Ensembles have been introduced by R. Kozhan several years ago. For a fixed N and beta>0 the joint probability density of N complex eigenvalues  was shown to have a form of a nontrivially interacting two-component Dyson-type Coulomb plasma in the lower half of the complex plane.  The problem of extracting the eigenvalue/eigenvector correlations in the large-N limit, including the simplest one-point eigenvalue density, remained outstanding beyond beta=2 where the plasma was asymptotically determinantal. 

We show how the computation of the mean eigenvalue density and the associated left-right non-orthogonal eigenvectors can be reduced to a certain correlation function of products and ratios of characteristic polynomials of  the underlying beta-Hermite Ensemble.  For the classical beta=1,2,4 such objects can be explicitly evaluated but general case remains  outstanding.   Attempting to study the problem perturbatively we arrive at a similar, yet simpler correlation functions for which we conjecture explicit formulas for any \beta>0.  As a by-product, we also conjecture associated  "eigenvalue curvatures'' distribution  characterizing the lowest-order shift in real parts of eigenvalues induced by the same non-Hermitian perturbation.  Numerical simulations show reasonable agreement to the conjectured expressions for a few non-classical values  of beta. The presentation is based on joint works with Mohammed Osman and Rashel Tublin.

Fyodorov-2022-06-22