On the almost eigenvectors of random regular graphs

Backhausz, Ágnes; Szegedy, Balázs

doi:10.1214/18-aop1294

Cited by 36 publications

(59 citation statements)

References 50 publications

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“…We also mention that in [3] the blow-ups of the star-edge inequality (2) were proved for a broader class of invariant processes that were called typical processes. These blow-up inequalities played a central role in the proof of the main result of that paper.…”

Section: New Inequalities For Aut(t D )-Factorsmentioning

confidence: 99%

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Entropy inequalities for factors of IID

2018

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This paper is concerned with certain invariant random processes (called factors of IID) on infinite trees. Given such a process, one can assign entropies to different finite subgraphs of the tree. There are linear inequalities between these entropies that hold for any factor of IID process (e.g. "edge versus vertex" or "star versus edge"). These inequalities turned out to be very useful: they have several applications already, the most recent one is the Backhausz-Szegedy result on the eigenvectors of random regular graphs.We present new entropy inequalities in this paper. In fact, our approach provides a general "recipe" for how to find and prove such inequalities. Our key tool is a generalization of the edge-vertex inequality for a broader class of factor processes with fewer symmetries.2010 Mathematics Subject Classification. 37A35, 60K35, 37A50, 05E18.

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Section: New Inequalities For Aut(t D )-Factorsmentioning

confidence: 99%

“…The above inequalities played a central role in a couple of intriguing results recently: the Rahman-Virág result [17] about the maximal size of a factor of IID independent set on T d and the Backhausz-Szegedy result [3] on the "local statistics" of eigenvectors of random regular graphs.…”

Section: Introductionmentioning

confidence: 99%

Entropy inequalities for factors of IID

2018

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“…We would like to notice that delocalization properties of eigenvectors for various models have been a focus of active research. In the case of sparse matrices we refer to [12] for eigenvector statistics for Erdős-Rényi graphs and to [2,3,4,18] for delocalization properties of eigenvectors (and almost eigenvectors) of undirected regular graphs. In the non-Hermitian setting (relevant to our present work) we refer to [52,53].…”

Section: Introductionmentioning

confidence: 99%

Structure of eigenvectors of random regular digraphs

Litvak

Lytova

Tikhomirov

et al. 2019

Trans. Amer. Math. Soc.

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Let d and n be integers satisfying C ≤ d ≤ exp(c √ ln n) for some universal constants c, C > 0, and let z ∈ C. Denote by M the adjacency matrix of a random d-regular directed graph on n vertices. In this paper, we study the structure of the kernel of submatrices of M − z Id, formed by removing a subset of rows. We show that with large probability the kernel consists of two non-intersecting types of vectors, which we call very steep and gradual with many levels. As a corollary, we show, in particular, that every eigenvector of M , except for constant multiples of (1, 1, . . . , 1), possesses a weak delocalization property: its level sets have cardinality less than Cn ln 2 d/ ln n. For a large constant d this provides a principally new structural information on eigenvectors, implying that the number of their level sets grows to infinity with n. As a key technical ingredient of our proofs we introduce a decomposition of C n into vectors of different degrees of "structuredness," which is an alternative to the decomposition based on the least common denominator in the regime when the underlying random matrix is very sparse.

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“…For a more precise statement see Theorem 1.2 in [BHY]. In another line of work, Backhausz and Szegedy [BS16] establish Gaussian behavior of the entry distribution of eigenvectors of random regular graphs by studying factors of i.i.d. processes on the regular infinite tree.…”

Section: Introductionmentioning

confidence: 99%

On Non-localization of Eigenvectors of High Girth Graphs

Ganguly

Srivastava

2019

International Mathematics Research Notices

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We prove improved bounds on how localized an eigenvector of a high girth regular graph can be, and present examples showing that these bounds are close to sharp. This study was initiated by Brooks and Lindenstrauss [BL13] who relied on the observation that certain suitably normalized averaging operators on high girth graphs are hyper-contractive and can be used to approximate projectors onto the eigenspaces of such graphs. Informally, their delocalization result in the contrapositive states that for any ε ∈ (0, 1) and positive integer k, if a (d + 1)−regular graph has an eigenvector which supports ε fraction of the 2 2 mass on a subset of k vertices, then the graph must have a cycle of sizeÕ(log d (k)/ε 2 ), suppressing logarithmic terms in 1/ . In this paper, we improve the upper bound toÕ(log d (k)/ε) and present a construction showing a lower bound of Ω(log d (k)/ε). Our construction is probabilistic and involves gluing together a pair of trees while maintaining high girth as well as control on the eigenvectors and could be of independent interest.

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On the almost eigenvectors of random regular graphs

Cited by 36 publications

References 50 publications

Entropy inequalities for factors of IID

Entropy inequalities for factors of IID

Structure of eigenvectors of random regular digraphs

On Non-localization of Eigenvectors of High Girth Graphs

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