Annales Henri Poincaré

, Volume 16, Issue 9, pp 1969–2003 | Cite as

Resonances and Partial Delocalization on the Complete Graph

  • Michael AizenmanEmail author
  • Mira Shamis
  • Simone Warzel


Random operators may acquire extended states formed from a multitude of mutually resonating local quasi-modes. This mechanics is explored here in the context of the random Schrödinger operator on the complete graph. The operator exhibits local quasi-modes mixed through a single channel. While most of its spectrum consists of localized eigenfunctions, under appropriate conditions it includes also bands of states which are delocalized in the \({\ell^{1}}\)-though not in \({\ell^{2}}\)-sense, where the eigenvalues have the statistics of Šeba spectra. The analysis proceeds through some general observations on the scaling limits of random functions in the Herglotz–Pick class. The results are in agreement with a heuristic condition for the emergence of resonant delocalization, which is stated in terms of the tunneling amplitude among quasi-modes.


Point Process Complete Graph Random Function Extended State Random Operator 
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© Springer Basel 2014

Authors and Affiliations

  1. 1.Departments of Physics and MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Mathematics DepartmentPrinceton UniversityPrincetonUSA
  3. 3.Zentrum MathematikTU MünchenGarchingGermany

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