Annales Henri Poincaré

, Volume 16, Issue 8, pp 1779–1806 | Cite as

Poisson Eigenvalue Statistics for Random Schrödinger Operators on Regular Graphs

  • Leander GeisingerEmail author


For random operators it is conjectured that spectral properties of an infinite-volume operator are related to the distribution of spectral gaps of finite-volume approximations. In particular, localization and pure point spectrum in infinite volume is expected to correspond to Poisson eigenvalue statistics. Motivated by results on the Anderson model on the infinite tree we consider random Schrödinger operators on finite regular graphs. We study local spectral statistics: we analyze the number of eigenvalues in intervals with length comparable to the inverse of the number of vertices of the graph, in the limit where this number tends to infinity. We show that the random point process generated by the rescaled eigenvalues converges in certain spectral regimes of localization to a Poisson process. The corresponding result on the lattice was proved by Minami. However, due to the geometric structure of regular graphs the known methods turn out to be difficult to adapt. Therefore, we develop a new approach based on direct comparison of eigenvectors.


Regular Graph Anderson Model Random Operator Bethe Lattice Pure Point Spectrum 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer Basel 2014

Authors and Affiliations

  1. 1.Department of PhysicsPrinceton UniversityPrincetonUSA

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