Outliers in spectrum of sparse Wigner matrices

In this paper, we study the effect of sparsity on the appearance of outliers in the semi‐circular law. Let (Wn)n=1∞ be a sequence of random symmetric matrices such that each Wn is n × n with i.i.d. entries above and on the main diagonal equidistributed with the product bnξ, where ξ is a real centere...

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Bibliographic Details
Published in:Random structures & algorithms Vol. 58; no. 3; pp. 517 - 605
Main Authors: Tikhomirov, Konstantin, Youssef, Pierre
Format: Journal Article
Language:English
Published: New York John Wiley & Sons, Inc 01.05.2021
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Subjects:
BBP phase transition
Convergence
Eigenvalues
Erdos‐Renyi graph
Mathematical analysis
Mathematics
Matrix methods
moment method
outliers
Phase transitions
Probability
Random variables
Sparse matrices
Sparsity
spectral gap
ISSN:1042-9832, 1098-2418
Online Access:Get full text
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Summary:In this paper, we study the effect of sparsity on the appearance of outliers in the semi‐circular law. Let (Wn)n=1∞ be a sequence of random symmetric matrices such that each Wn is n × n with i.i.d. entries above and on the main diagonal equidistributed with the product bnξ, where ξ is a real centered uniformly bounded random variable of unit variance and bn is an independent Bernoulli random variable with a probability of success pn. Assuming that limn→∞npn=∞, we show that for the random sequence (ρn)n=1∞ given by ρn:=θn+npnθn,θn:=max(maxi≤n‖rowi(Wn)‖22−npn,npn), the ratio ‖Wn‖ρn converges to one in probability. A noncentered counterpart of the theorem allows to obtain asymptotic expressions for eigenvalues of the Erdős–Renyi graphs, which were unknown in the regime npn=Θ(logn). In particular, denoting by An the adjacency matrix of the Erdős–Renyi graph 𝒢(n,pn) and by λ|k|(An) its kth largest (by the absolute value) eigenvalue, under the assumptions limn→∞npn=∞ and limn→∞pn=0 we have (1) (No non‐trivial outliers): if liminfnpnlogn≥1log(4/e) then for any fixed k ≥ 2, |λ|k|(An)|2npn converges to 1 in probability; and (2) (Outliers): if limsupnpnlogn<1log(4/e) then there is ε > 0 such that for any k∈ℕ, we have limn→∞ℙ|λ|k|(An)|2npn>1+ε=1. On a conceptual level, our result reveals similarities in appearance of outliers in spectrum of sparse matrices and the so‐called BBP phase transition phenomenon in deformed Wigner matrices.
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ISSN:1042-9832
1098-2418
DOI:10.1002/rsa.20982