Level curvature distribution: from bulk to the soft edge of random Hermitian matrices

TL;DR

This paper investigates the distribution of level curvatures near the spectral edge of Hermitian random matrices, extending known bulk results to the soft edge using asymptotic analysis of orthogonal polynomial kernels.

Contribution

It introduces a method to analyze level curvature distributions at the spectral edge of Hermitian matrices, connecting to orthogonal polynomial asymptotics and applicable to invariant ensembles.

Findings

Derived level curvature distribution near the spectral edge.

Extended bulk distribution results to the soft edge.

Method based on asymptotic analysis of Hermite polynomial kernels.

Abstract

Level curvature is a measure of sensitivity of energy levels of a disordered/chaotic system to perturbations. In the bulk of the spectrum Random Matrix Theory predicts the probability distributions of level curvatures to be given by Zakrzewski-Delande expressions [F. von Oppen {\it Phys. Rev. Lett.} {\bf 73} 798 (1994) & {\it Phys.Rev. E} {\bf 51} 2647 (1995); Y.V. Fyodorov and H.-J. Sommers {\it Z.Phys.B} {\bf 99} 123 (1995)]. Motivated by growing interest in statistics of extreme (maximal or minimal) eigenvalues of disordered systems of various nature, it is natural to ask about the associated level curvatures. I show how calculating the distribution for the curvatures of extreme eigenvalues in GUE ensemble can be reduced to studying asymptotic behaviour of orthogonal polynomials appearing in the recent work C. Nadal and S. N. Majumdar {\it J. Stat. Mech.} {\bf 2011} P04001 (2011). The corresponding asymptotic analysis being yet outstanding, I instead will discuss solution of a related, but somewhat simpler problem of calculating the level curvature distribution averaged over all the levels in a spectral window close to the edge of the semicircle. The method is based on asymptotic analysis of kernels associated with Hermite polynomials and their Cauchy transforms, and is straightforwardly extendable to any rotationally-invariant ensemble of random matrices.