Two short pieces around the Wigner problem

Abstract

We revisit the classic Wigner semi-circle from two different angles. One consists in studying the Stieltjes transform directly on the real axis, which does not converge to a fixed value but follows a Cauchy distribution that depends on the local eigenvalue density. This result was recently proven by Aizenman \& Warzel for a wide class of eigenvalue distributions. We shed new light onto their result using a Coulomb gas method. The second angle is to derive a Langevin equation for the full (matrix) resolvent, extending Dyson's Brownian motion framework. The full matrix structure of this equation allows one to recover known results on the overlaps between the eigenvectors of a fixed matrix and its noisy counterpart.

Some minor misprints corrected. To appear in J. Phys. A, special issue: Random Matrices, the first 90 years

References in corpus (6)

• Cleaning large correlation matrices: tools from random matrix theory
• Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices
• Rotational invariant estimator for general noisy matrices
• Statistics of Impedance, Local Density of States, and Reflection in Quantum Chaotic Systems with Absorption
• Distribution of spectral linear statistics on random matrices beyond the large deviation function -- Wigner time delay in multichannel disordered wires
• The eigenvectors of Gaussian matrices with an external source