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.

Anderson transition on the Bethe lattice: an approach with real energies

“…which gives Λ = µ (s * ). (58) Notice that, due to the symmetry µ(s) = µ(1−s) we have, for s * 1 and W W erg ,…”

“…This agrees with numerical results, as shown in figure 1, where the distribution of P in ( 12) is compared with numerical solutions of equation ( 2) on an ensemble of RRGs. The fact that (for random operators) the diagonal matrix elements of the resolvent evaluated on the real axis are Cauchy variables is a property that holds in surprising generality [57,58], irrespectively of the statistics of the matrix eigenvalues and of the possible strong correlations between them, as exemplified by Coulomb gas models [58][59][60]. Moreover, the fact that the numerics on RRG is so accurately described by a cavity equation means that the correlations along closed paths of length ln K N which loop around the RRG are irrelevant for this quantity, and that the G k 's can be considered as independent, identically distributed variables.…”