Chiral Random Matrix Models: A Novel Intermediate Asymptotic Regime

TL;DR

This paper investigates the asymptotic behavior of the Chiral Random Matrix Model, extending universality to non-polynomial potentials and exploring correlations relevant for mesoscopic systems and structural glasses.

Contribution

It introduces a new asymptotic ansatz for the Chiral Random Matrix Model, extending universality to include non-polynomial potentials using saddle point techniques.

Findings

Density-density correlators match double well models

Correlators are sensitive to matrix size N (odd/even)

Results applicable to mesoscopic systems with eigenvalue gaps

Abstract

The Chiral Random Matrix Model or the Gaussian Penner Model (generalized Laguerre ensemble) is re-examined in the light of the results which have been found in double well matrix models [D97,BD99] and subtleties discovered in the single well matrix models [BH99]. The orthogonal polynomial method is used to extend the universality to include non-polynomial potentials. The new asymptotic ansatz is derived (different from Szego's result) using saddle point techniques. The density-density correlators are the same as that found for the double well models ref. [BD99] (there the results have been derived for arbitrary potentials). In the smoothed large N limit they are sensitive to odd and even N where N is the size of the matrix [BD99]. This is a more realistic random matrix model of mesoscopic systems with density of eigenvalues with gaps. The eigenvalues see a brick-wall potential at the origin. This would correspond to sharp edges in a real mesoscopic system or a reflecting boundary. Hence the results for the two-point density-density correlation function may be useful in finding one eigenvalue effects in experiments in mesoscopic systems or small metallic grains. These results may also be relevant for studies of structural glasses as described in ref. [D02].