Anderson localization for Bernoulli and other singular potentials

Rene Carmona, Abel Klein, Fabio Martinelli

Research output: Contribution to journalArticlepeer-review

185 Scopus citations


We prove exponential localization in the Anderson model under very weak assumptions on the potential distribution. In one dimension we allow any measure which is not concentrated on a single point and possesses some finite moment. In particular this solves the longstanding problem of localization for Bernoulli potentials (i.e., potentials that take only two values). In dimensions greater than one we prove localization at high disorder for potentials with Hölder continuous distributions and for bounded potentials whose distribution is a convex combination of a Hölder continuous distribution with high disorder and an arbitrary distribution. These include potentials with singular distributions. We also show that for certain Bernoulli potentials in one dimension the integrated density of states has a nontrivial singular component.

Original languageEnglish (US)
Pages (from-to)41-66
Number of pages26
JournalCommunications In Mathematical Physics
Issue number1
StatePublished - Mar 1987
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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