Non-localization of eigenfunctions on large regular graphs

Abstract

We give a delocalization estimate for eigenfunctions of the discrete Laplacian on large d + 1-regular graphs, showing that any subset of the graph supporting ǫ of the L 2 mass of an eigenfunction must be large. For graphs satisfying a mild girth-like condition, this bound will be exponential in the size of the graph.

Random Band Matrices

“…and ψ k L 2 (µ) = 1. This quantum unique ergodicity (QUE) notion strengthens the quantum ergodicity proved in [35,90,104], defined by an additional averaging on k and proved for a wide class of manifolds and deterministic regular graphs [5] (see also [30]). QUE was rigorously proved for arithmetic surfaces, [62,63,76].…”

Local Semicircle Law for Random Regular Graphs

“…Let ξ log ξ ≫ (log N ) 2 and D ≫ ξ 2 . Then, with probability at least 1 − e −ξ log ξ , 12) simultaneously for all z ∈ C + such that η ≫ ξ 2 /N .…”

“…In particular, it is known that the spectral measure of any sequence of graphs converging locally to the d -regular tree converges to the Kesten-McKay law; see, for instance, [8]. Moreover, in [11], under an assumption on the number of small cycles (corresponding approximately to a locally treelike structure and satisfied with high probability by random regular graphs), it is proved that eigenvectors cannot be localized in the following sense: if for some`2-normalized eigenvector v D .v i / N i D1 a set B 1; N satisfies P i 2B jv i j 2 > " > 0, then jBj > N ı with high probability for some small ı / " 2 . In comparision, for a random d -regular graph with d > .log N / 4 , Corollary 1.2 implies that if a set B has`2-mass " > 0, then jBj > "N.log N / 4 with high probability, which is optimal up to the power of the logarithmic correction.…”

“…In particular, it is known that the spectral measure of any sequence of graphs converging locally to the d-regular tree converges to the Kesten-McKay law; see for instance [9]. Moreover, in [12], under an assumption on the number of small cycles (corresponding approximately to a locally tree-like structure and satisfied with high probability by random regular graphs), it is proved that eigenvectors cannot be localized in the following sense: if for some…”

$$L^p$$ Norms and Support of Eigenfunctions on Graphs

“…It is shown in [8,22,7] that most eigenfunctions of such graphs are uniformly distributed in a certain sense, a property known as quantum ergodicity. Lower bounds on the support were provided in [20], where it is shown more generally that the eigenfunctions cannot concentrate on small sets. The recent paper [21] provides norm estimates which read as ψ λ p 1 (log q |G|) 1/2 for all p > 2, for generic regular graphs.…”