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Quantum ergodicity on graphs: from spectral to spatial delocalization. (English) Zbl 1423.58021
Let $$G=(V,E)$$ be a finite graph. Fix an interval $$I\subset\mathbb{R}$$. The authors examine the extent to which the eigenfunctions of the adjacency matrix are localized or delocalized. One says that one has spectral delocalication if there is purely absolutely continuous spectrum in $$I$$. The authors consider a sequence of finite graphs which have discrete Schrödinger operators which have a local weak limit model which is a random rooted tree which has a random discrete Schrödinger model and study when quantum ergodicity pertains; this is a form of spatial delocalization of the eigenvalues of the finite graphs which aproximate the weak model. Roughly speaking, this implies that the eigenfunctions become equi-distributed in phase space. The results of the paper apply to graphs which converge to the Anderson model of a regular tree.
The main result of the paper can be roughly stated as follows: “If a large finite system is close in the Benjamini-Schramm topology to an infinite system having a purely absolutely continuous spectrum in an interval I, then the eigenfunctions with eigenvalues lying in I of the finite system satisfy quantum ergodicity.”
Section 1 is an introduction to the subject. Section 2 provides some basic identities. Section 3 discusses the non-backtracking quantum variance. Section 4 provides a bound on the non-backtracking quantum variance. Section 5 presents an invariance property of the quantum variance. Section 6 involves a stationary Markov chain. Section 7 deals with the spectral gap and mixing. Section 8 treats transition matrices with phases. Section 9 uses these results to establish one of the results of the paper. In Section 10, one returns to the original eigenfunctions by showing it suffices to consider the non-backtracking quantum variance to prove quantum ergodicity. In Appendix A, basic facts concerning the Benjamini-Schramm topology are given.

##### MSC:
 58J51 Relations between spectral theory and ergodic theory, e.g., quantum unique ergodicity
##### Keywords:
quantum ergodicity; large graphs; delocalization
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