A necessary and sufficient condition for edge universality of Wigner matrices

Lee, Ji Oon; Yin, Jun

doi:10.1215/00127094-2414767

Cited by 98 publications

(140 citation statements)

References 52 publications

Supporting

Mentioning

135

Contrasting

Order By: Relevance

“…for some δ > 0, see e.g. [7], [5], [20], [13], [14], [15], [18] and [17]. In particular, the result of [17] implies that (1.4) holds with α(n) ≡ 1.…”

Section: Introduction and Main Resultsmentioning

confidence: 98%

Local Semicircle Law Under Fourth Moment Condition

2019

View full text Add to dashboard Cite

We consider a random symmetric matrix X = [X jk ] n j,k=1 with upper triangular entries being independent random variables with mean zero and unit variance. Assuming that max jk E |X jk | 4+δ < ∞, δ > 0, it was proved in [17] that with high probability the typical distance between the Stieltjes transforms mn(z), z = u + iv, of the empirical spectral distribution (ESD) and the Stieltjes transforms msc(z) of the semicircle law is of order (nv) −1 log n. The aim of this paper is to remove δ > 0 and show that this result still holds if we assume that max jk E |X jk | 4 < ∞. We also discuss applications to the rate of convergence of the ESD to the semicircle law in the Kolmogorov distance, rates of localization of the eigenvalues around the classical positions and rates of delocalization of eigenvectors.

show abstract

“…for some δ > 0, see e.g. [7], [5], [20], [13], [14], [15], [18] and [17]. In particular, the result of [17] implies that (1.4) holds with α(n) ≡ 1.…”

Section: Introduction and Main Resultsmentioning

confidence: 98%

Local Semicircle Law Under Fourth Moment Condition

2019

View full text Add to dashboard Cite

show abstract

“…() have removed the symmetry condition and established the edge universality result for general Wigner ensembles. Further Lee and Yin () showed a necessary and sufficient condition for having the limiting Tracy–Widom law, which shows that

n^{2 / 3} false(λ_{1} - 2 false)

converges weakly to

{TW}_{1}

also. If we know the true p , it would be easy to frame a hypothesis test which accepts or rejects the null hypothesis that a network is generated from an Erdős–Rényi graph.…”

Section: The Hypothesis Testmentioning

confidence: 97%

“…the network is generated from an Erdős–Rényi

G_{n, p}

‐graph, where n denotes the number of nodes and p denotes the probability of linkage between a pair of nodes. Existing literature (Lee and Yin, ) can be used to show that this largest eigenvalue asymptotically has the Tracy–Widom distribution. Using recent theoretical results from random‐matrix theory, we show that this limit also holds for our statistic, when the probability of an edge p is unknown, and the centring and scaling are done using an estimate of p .…”

Section: Introductionmentioning

confidence: 99%

Hypothesis Testing for Automated Community Detection in Networks

Bickel

Sarkar

2015

Journal of the Royal Statistical Society Series B: Statistical Methodology

159

145

View full text Add to dashboard Cite

Community detection in networks is a key exploratory tool with applications in a diverse set of areas, ranging from finding communities in social and biological networks to identifying link farms in the World Wide Web. The problem of finding communities or clusters in a network has received much attention from statistics, physics and computer science. However, most clustering algorithms assume knowledge of the number of clusters k. In this paper we propose to automatically determine k in a graph generated from a Stochastic Blockmodel. Our main contribution is twofold; first, we theoretically establish the limiting distribution of the principal eigenvalue of the suitably centered and scaled adjacency matrix, and use that distribution for our hypothesis test. Secondly, we use this test to design a recursive bipartitioning algorithm. Using quantifiable classification tasks on real world networks with ground truth, we show that our algorithm outperforms existing probabilistic models for learning overlapping clusters, and on unlabeled networks, we show that we uncover nested community structure.

show abstract

“…Universality at the edge of the spectrum of sample covariance and correlation matrices has been studied by Feldheim and Sodin (2010), Bao et al (2012) and Pillai and Jin (2012). Lee and Yin (2012) established necessary and sufficient conditions for edge universality of a Wigner matrix. Large deviations of the extreme eigenvalues have been studied by Benaych-Georges et al (2012).…”

Section: Universalitymentioning

confidence: 99%

Random matrix theory in statistics: A review

Paul

Aue

2014

Journal of Statistical Planning and Inference

165

115

View full text Add to dashboard Cite

b s t r a c tWe give an overview of random matrix theory (RMT) with the objective of highlighting the results and concepts that have a growing impact in the formulation and inference of statistical models and methodologies. This paper focuses on a number of application areas especially within the field of high-dimensional statistics and describes how the development of the theory and practice in high-dimensional statistical inference has been influenced by the corresponding developments in the field of RMT.

show abstract

A necessary and sufficient condition for edge universality of Wigner matrices

Cited by 98 publications

References 52 publications

Local Semicircle Law Under Fourth Moment Condition

Local Semicircle Law Under Fourth Moment Condition

Hypothesis Testing for Automated Community Detection in Networks

Random matrix theory in statistics: A review

Contact Info

Product

Resources

About