On the ubiquity of the Cauchy distribution in spectral problems

Michael Aizenman, Simone Warzel

Research output: Contribution to journalArticlepeer-review

10 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)61-87
Number of pages27
JournalProbability Theory and Related Fields
Issue number1-2
StatePublished - Oct 1 2015

All Science Journal Classification (ASJC) codes

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty


  • Primary 60E99
  • Secondary 15B52


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