Level Curvature Distribution: from Bulk to the Soft Edge of Random Hermitian Matrices

• Authors: Y. Fyodorov
• Languages of publication: EN

Abstracts

EN

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 the Zakrzewski-Delande expressions. 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. We show how calculating the distribution for the curvatures of extreme eigenvalues in Gaussian unitary ensemble can be reduced to studying asymptotic behaviour of orthogonal polynomials appearing in the recent work of Nadal and Majumdar. The corresponding asymptotic analysis being yet outstanding, we 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.

Keywords

EN

05.40.-a

Discipline

• 05.40.-a: Fluctuation phenomena, random processes, noise, and Brownian motion(for fluctuations in superconductivity, see 74.40.-n; for statistical theory and fluctuations in nuclear reactions, see 24.60.-k; for fluctuations in plasma, see 52.25.Gj; for nonlinear dynamics and chaos, see 05.45.-a)

• Publisher: Institute of Physics, Polish Academy of Sciences
• Journal: Acta Physica Polonica A
• Year: 2011
• Volume: 120
• Issue: 6A
• Pages: A-100-A-113

Dates

published

2011-12

Contributors

author

Y. Fyodorov

• School of Mathematical Sciences, University of Nottingham, Nottingham NG72RD, England, UK

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Identifiers

DOI

10.12693/APhysPolA.120.A-100