Francisco Hoecker-Escuti, Christoph Schumacher: The Anderson model on the Bethe lattice: Lifshitz Tails

Author(s):

Francisco Hoecker-Escuti
Christoph Schumacher

Title:

Francisco Hoecker-Escuti, Christoph Schumacher: The Anderson model on the Bethe lattice: Lifshitz Tails

Electronic source:

application/pdf

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 17, 2014

Mathematics Subject Classification:

    60H25 []
    35J10 []

Abstract:

This paper is devoted to the study of the (discrete) Anderson Hamiltonian on the Bethe lattice, which is an infinite tree with constant vertex degree. The Hamiltonian we study corresponds to the sum of the graph Laplacian and a diagonal operator with non-negative bounded, i.i.d. random coefficients on its diagonal. We study in particular the asymptotic behavior of the integrated density of states near the bottom of the spectrum. More precisely, under a natural condition on the random variables, we prove the conjectured double-exponential Lifschitz tail with exponent 1/2. We study the Laplace transform of the density of states. It is related to the solution of the parabolic Anderson problem on the tree. These estimates are linked to the asymptotic behavior of the ground state energy of the Anderson Hamiltonian restricted to trees of finite length. The proofs make use of Tauberian theorems, a discrete Feynman--Kac formula, a discrete IMS localization formula, the spectral theory of the free Laplacian on finite rooted trees, an uncertainty principle for low-energy states, an epsilon-net argument and the standard concentration inequalities.

Keywords:

Language:

English

Publication time:

10/2014