Quantum ergodicity on graphs: From spectral to spatial delocalization

Abstract

We prove a quantum-ergodicity theorem on large graphs, for eigenfunctions of Schrödinger operators in a very general setting. We consider a sequence of finite graphs endowed with discrete Schrödinger operators, assumed to have a local weak limit. We assume that our graphs have few short loops, in other words that the limit model is a random rooted tree endowed with a random discrete Schrödinger operator. We show that absolutely continuous spectrum for the infinite model, reinforced by a good control of the moments of the Green function, imply "quantum ergodicity", a form of spatial delocalization for eigenfunctions of the finite graphs approximating the tree. This roughly says that the eigenfunctions become equidistributed in phase space. Our result applies in particular to graphs converging to the Anderson model on a regular tree, in the régime of extended states studied by Klein and Aizenman-Warzel.2010 Mathematics Subject Classification. Primary 58J51. Secondary 60B20, 81Q10.

Quantum ergodicity for large equilateral quantum graphs

“…These assumptions were also needed to prove quantum ergodicity for discrete graphs in . They are known to be ‘generic’, in the sense that a regular graph picked at random will typically satisfy these assumptions.…”

“…where ρ GN (x) is the largest ρ such that the sub-graph contained in a ball of radius ρ centred at x has no closed cycles. These assumptions were also needed to prove quantum ergodicity for discrete graphs in [2,4]. They are known to be 'generic', in the sense that a regular graph picked at random will typically satisfy these assumptions.…”

“…One cannot apply the results of directly to the operators appearing in the proof. Still, we follow the general arguments of [, Section 4], and we were able to simplify many ideas for the specific framework of our paper. Now that we finally have Hilbert–Schmidt norms of discrete observables, we can follow the scheme of to conclude the proof.…”

“…We then prove our main result in Section . As in the case of combinatorial graphs , one can also ask about the behaviour of the eigenfunction correlators . For this purpose, we provide a more general quantum ergodicity result, Theorem , involving integral operators.…”

“…We then prove our main result in Section 4. As in the case of combinatorial graphs [2][3][4], one can also ask about the behaviour of the eigenfunction correlators ψ…”