TY - JOUR

T1 - Quantum ergodicity on graphs

T2 - From spectral to spatial delocalization

AU - Anantharaman, Nalini

AU - Sabri, Mostafa

N1 - Funding Information:
Acknowledgements. This material is based upon work supported by the Agence Nationale de la Recherche under grant No.ANR-13-BS01-0007-01, by the Labex IRMIA and the Institute of Advance Study of Universitéde Strasbourg, and by Institut Universitaire de France.
Publisher Copyright:
© 2019 Annals of Mathematics.

PY - 2019/5/1

Y1 - 2019/5/1

N2 - 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 an 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 regime of extended states studied by Klein and Aizenman-Warzel.

AB - 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 an 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 regime of extended states studied by Klein and Aizenman-Warzel.

KW - Delocalization

KW - Large graphs

KW - Quantum ergodicity

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U2 - 10.4007/annals.2019.189.3.3

DO - 10.4007/annals.2019.189.3.3

M3 - Article

AN - SCOPUS:85066923527

SN - 0003-486X

VL - 189

SP - 753

EP - 835

JO - Annals of Mathematics

JF - Annals of Mathematics

IS - 3

ER -