2014/06/06 by Leander Geisinger, Geisinger, Leander
Mathematics · #05C80 #35P20 #60B20 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Probability (math.PR) #Random Matrices and Applications #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1406.1608
openalex publication_date 2014/06/06 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28
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 about 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.