Level Curvature Distribution: from Bulk to the Soft Edge of Random Hermitian Matrices
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 ZakrzewskiDelande 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 rotationallyinvariant ensemble of random matrices.
 Publication:

Acta Physica Polonica A
 Pub Date:
 December 2011
 DOI:
 10.12693/APhysPolA.120.A100
 arXiv:
 arXiv:1108.0950
 Bibcode:
 2011AcPPA.120..100F
 Keywords:

 Mathematical Physics;
 Condensed Matter  Disordered Systems and Neural Networks
 EPrint:
 The text is based on the presentation at the 5th Workshop on Quantum Chaos and Localization Phenomena, May 2022, 2011, Warsaw, Poland