On the ubiquity of the Cauchy distribution in spectral problems

Abstract

We consider the distribution of the values at real points of random functions which belong to the Herglotz-Pick (HP) class of analytic mappings of the upper half plane into itself. It is shown that under mild stationarity assumptions the individual values of HP functions with singular spectra have a Cauchy type distribution. The statement applies to the diagonal matrix elements of random operators, and holds regardless of the presence or not of level repulsion, i.e. applies to both random matrix and Poisson-type spectra.

On the critical points of random matrix characteristic polynomials and of the Riemann ξ-function

“…, if the sums are properly regularised, whereas the first relation, (20) requires a deterministic correction depending on E; see Sections 2.3 and 2.2 (relying on the works [34] and [1], respectively). These properties imply that the critical points depend quasi-locally, so to speak, on the zeros / eigenvalues.…”

“…The two limits exist and coincide according to a general criterion of [1], and W(z) is a random element of the Nevanlinna class. Also note that −W(z) is the logarithmic derivative of the function Φ(z) from (4).…”

The Limiting Characteristic Polynomial of Classical Random Matrix Ensembles

“…Much of the information that has traditionally been of interest can be succinctly summarized within this new framework and furthermore new questions are brought to the surface by the change of focus. Papers making use of this perspective include [AW15,CNN17,Sod17].…”

Two short pieces around the Wigner problem

“…Most of the arguments of section II A only rely on the fact that the elements of W are independent, identically distributed random random variables with a finite second moment, but not necessarily Gaussian. If the second moment is finite, we expect that all the above results will generalize, since they only rely on local properties of the spectrum (see [4,10]). Let us remind the reader how such arguments must be adapted to the case where the matrix elements of W have a diverging second moment, or more precisely when:…”

“…In this case, the Cauchy distribution for g was actually derived by Y. Fyodorov and collaborators for the GOE and GUE ensembles, using rather specific Random Matrix Theory techniques [1][2][3]. It was recently proven by Aizenman & Warzel [4] that the Cauchy distribution is in fact super-universal and holds not only for all Coulomb gas models, for arbitrary values of β, but in fact for a much wider class of point processes on the real axis. The gist of the argument of Aizenman & Warzel is summarized in Appendix A.…”