JournalsjstVol. 5, No. 4pp. 783–827

Convergence of the density of states and delocalization of eigenvectors on random regular graphs

  • Leander Geisinger

    Universität Stuttgart, Germany
Convergence of the density of states and delocalization of eigenvectors on random regular graphs cover
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Abstract

Consider a random regular graph of fixed degree dd with nn vertices. We study spectral properties of the adjacency matrix and of random Schrödinger operators on such a graph as nn tends to infinity.

We prove that the integrated density of states on the graph converges to the integrated density of states on the infinite regular tree and we give uniform bounds on the rate of convergence. This allows to estimate the number of eigenvalues in intervals of size comparable to logd11(n)\log_{d-1}^{-1}(n). Based on related estimates for the Green function we derive results about delocalization of eigenvectors.

Cite this article

Leander Geisinger, Convergence of the density of states and delocalization of eigenvectors on random regular graphs. J. Spectr. Theory 5 (2015), no. 4, pp. 783–827

DOI 10.4171/JST/114