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Anantharaman, Nalini; Le Masson, Etienne

Quantum ergodicity on large regular graphs. (English) Zbl 1386.58015

Duke Math. J. 164, No. 4, 723-765 (2015).
Let \((G_n)\) be a sequence of \((q+1)\)-regular graphs, \(G_n=(V_n,E_n)\) with \(V_n=\{1,\dots,n\}\). Assume that \((G_n)\) is a family of expanders and that it converges to a tree in the sense of Benjamini and Schramm. Fix \(s_0\in(0,\tau)\), and let \(I_n=[s_0-\delta_n,s_0+\delta_n]\). Call \((s_1^{(n)},\dots,s_n^{(n)})\) the spectral parameters associated with the eigenvalues of \(M=\Delta+I\) with \(\Delta\) the discrete Laplacian, and \((\psi_1^{(n)},\dots,\psi_n^{(n)})\) a corresponding orthonormal eigenbasis.
Let \(N(I_n,G_n)=|\{j\in\{1,\dots,n\}\;|\;s_j^{(n)}\in I_n\}|\) be the number of eigenvalues in \(I_n\). Denote by \(a_n:\;V_n\to \mathbb C\) a sequence of functions such that \[ \sum_{x\in V_n}a_n(x)=0,\qquad \sup_x|a_n(x)|\leq 1. \]
The main theorem of this article asserts that \[ \lim_{n\to \infty} 1{N(I_n,G_n)}\sum_{s_j^{(n)}\in I_n}|\langle\psi_j^{(n)},a_n\psi_j^{(n)}\rangle|^2 =0. \] The authors also generalize it by replacing the function \(a\) with any finite range operator.
Reviewer: Shin-ya Koyama (Yokohama)

Cited in 2 Reviews
Cited in 42 Documents

MSC:

58J51 Relations between spectral theory and ergodic theory, e.g., quantum unique ergodicity
60B20 Random matrices (probabilistic aspects)
81Q50 Quantum chaos

Keywords:

large random graphs; Laplacian eigenfunctions; quantum ergodicity; semiclassical measures
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References:

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