Quantum ergodicity for the Anderson model on regular graphs

Abstract

We prove a result of delocalization for the Anderson model on the regular tree (Bethe lattice). When the disorder is weak, it is known that large parts of the spectrum are a.s. purely absolutely continuous, and that the dynamical transport is ballistic. In this work, we prove that in such AC regime, the eigenfunctions are also delocalized in space, in the sense that if we consider a sequence of regular graphs converging to the regular tree, then the eigenfunctions become asymptotically uniformly distributed. The precise result is a quantum ergodicity theorem.

References in corpus (3)

• The Canopy Graph and Level Statistics for Random Operators on Trees
• Level compressibility for the Anderson model on regular random graphs and the eigenvalue statistics in the extended phase
• Absolutely continuous spectrum implies ballistic transport for quantum particles in a random potential on tree graphs

Cited by in corpus (3)

• Anderson Localization on the Bethe Lattice using Cages and the Wegner Flow
• Absolutely Continuous Spectrum for Quantum Trees
• $L^p$ norms and support of eigenfunctions on graphs