On the ubiquity of the Cauchy distribution in spectral problems

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.