RANDOM MATRICES: THE CIRCULAR LAW

Let x be a complex random variable with mean zero and bounded variance σ2. Let Nn be a random matrix of order n with entries being i.i.d. copies of x. Let λ1, …, λn be the eigenvalues of [Formula: see text]. Define the empirical spectral distributionμn of Nn by the formula [Formula: see text] The following well-known conjecture has been open since the 1950's: Circular Law Conjecture: μn converges to the uniform distribution μ∞ over the unit disk as n tends to infinity. We prove this conjecture, with strong convergence, under the slightly stronger assumption that the (2 + η)th-moment of x is bounded, for any η > 0. Our method builds and improves upon earlier work of Girko, Bai, Götze–Tikhomirov, and Pan–Zhou, and also applies for sparse random matrices. The new key ingredient in the paper is a general result about the least singular value of random matrices, which was obtained using tools and ideas from additive combinatorics.

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Complex random matrices have no real eigenvalues

Random Matrices Theory and Application ◽

10.1142/s2010326317500149 ◽

2018 ◽

Vol 07 (01) ◽

pp. 1750014 ◽

Cited By ~ 2

Author(s):

Kyle Luh

Keyword(s):

Random Matrices ◽

Random Variables ◽

Principal Component ◽

Random Variable ◽

Singular Values ◽

Singular Value ◽

Small Ball ◽

Circular Law ◽

Least Singular Value ◽

Tail Bound

Let [Formula: see text] where [Formula: see text] are iid copies of a mean zero, variance one, subgaussian random variable. Let [Formula: see text] be an [Formula: see text] random matrix with entries that are iid copies of [Formula: see text]. We prove that there exists a [Formula: see text] such that the probability that [Formula: see text] has any real eigenvalues is less than [Formula: see text] where [Formula: see text] only depends on the subgaussian moment of [Formula: see text]. The bound is optimal up to the value of the constant [Formula: see text]. The principal component of the proof is an optimal tail bound on the least singular value of matrices of the form [Formula: see text] where [Formula: see text] is a deterministic complex matrix with the condition that [Formula: see text] for some constant [Formula: see text] depending on the subgaussian moment of [Formula: see text]. For this class of random variables, this result improves on the results of Pan–Zhou [Circular law, extreme singular values and potential theory, J. Multivariate Anal. 101(3) (2010) 645–656] and Rudelson–Vershynin [The Littlewood–Offord problem and invertibility of random matrices, Adv. Math. 218(2) (2008) 600–633]. In the proof of the tail bound, we develop an optimal small-ball probability bound for complex random variables that generalizes the Littlewood–Offord theory developed by Tao–Vu [From the Littlewood–Offord problem to the circular law: Universality of the spectral distribution of random matrices, Bull. Amer. Math. Soc.[Formula: see text]N.S.[Formula: see text] 46(3) (2009) 377–396; Inverse Littlewood–Offord theorems and the condition number of random discrete matrices, Ann. of Math.[Formula: see text] 169(2) (2009) 595–632] and Rudelson–Vershynin [The Littlewood–Offord problem and invertibility of random matrices, Adv. Math. 218(2) (2008) 600–633; Smallest singular value of a random rectangular matrix, Comm. Pure Appl. Math. 62(12) (2009) 1707–1739].

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The strong circular law: A combinatorial view

Random Matrices Theory and Application ◽

10.1142/s2010326321500313 ◽

2020 ◽

pp. 2150031 ◽

Cited By ~ 1

Author(s):

Vishesh Jain

Keyword(s):

Error Rate ◽

Random Matrix ◽

Random Variable ◽

Singular Value ◽

Operator Norm ◽

Additive Combinatorics ◽

Counting Problem ◽

Circular Law ◽

Matrix Formula ◽

Least Singular Value

Let [Formula: see text] be an [Formula: see text] complex random matrix, each of whose entries is an independent copy of a centered complex random variable [Formula: see text] with finite nonzero variance [Formula: see text]. The strong circular law, proved by Tao and Vu, states that almost surely, as [Formula: see text], the empirical spectral distribution of [Formula: see text] converges to the uniform distribution on the unit disc in [Formula: see text]. A crucial ingredient in the proof of Tao and Vu, which uses deep ideas from additive combinatorics, is controlling the lower tail of the least singular value of the random matrix [Formula: see text] (where [Formula: see text] is fixed) with failure probability that is inverse polynomial. In this paper, using a simple and novel approach (in particular, not using machinery from additive combinatorics or any net arguments), we show that for any fixed complex matrix [Formula: see text] with operator norm at most [Formula: see text] and for all [Formula: see text], [Formula: see text] where [Formula: see text] is the least singular value of [Formula: see text] and [Formula: see text] are positive absolute constants. Our result is optimal up to the constants [Formula: see text] and the inverse exponential-type error rate improves upon the inverse polynomial error rate due to Tao and Vu. Our proof relies on the solution to the so-called counting problem in inverse Littlewood–Offord theory, developed by Ferber, Luh, Samotij, and the author, a novel complex anti-concentration inequality, and a “rounding trick” based on controlling the [Formula: see text] operator norm of heavy-tailed random matrices.

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Universality of the least singular value for sparse random matrices

Electronic Journal of Probability ◽

10.1214/19-ejp269 ◽

2019 ◽

Vol 24 (0) ◽

Cited By ~ 1

Author(s):

Ziliang Che ◽

Patrick Lopatto

Keyword(s):

Random Matrices ◽

Singular Value ◽

Sparse Random Matrices ◽

Least Singular Value

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Least singular value, circular law, and Lindeberg exchange

Random Matrices - IAS/Park City Mathematics Series ◽

10.1090/pcms/026/10 ◽

2019 ◽

pp. 461-498

Author(s):

Terence Tao

Keyword(s):

Singular Value ◽

Circular Law ◽

Least Singular Value

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On minimal singular values of random matrices with correlated entries

Random Matrices Theory and Application ◽

10.1142/s2010326315500069 ◽

2015 ◽

Vol 04 (02) ◽

pp. 1550006 ◽

Cited By ~ 6

Author(s):

F. Götze ◽

A. Naumov ◽

A. Tikhomirov

Keyword(s):

Random Matrices ◽

Random Matrix ◽

Singular Values ◽

Singular Value ◽

The Matrix ◽

Class Of Matrices ◽

Least Singular Value ◽

Mutually Independent

Let X be a random matrix whose pairs of entries Xjk and Xkj are correlated and vectors (Xjk, Xkj), for 1 ≤ j < k ≤ n, are mutually independent. Assume that the diagonal entries are independent from off-diagonal entries as well. We assume that [Formula: see text], for any j, k = 1, …, n and 𝔼 XjkXkj = ρ for 1 ≤ j < k ≤ n. Let Mn be a non-random n × n matrix with ‖Mn‖ ≤ KnQ, for some positive constants K > 0 and Q ≥ 0. Let sn(X + Mn) denote the least singular value of the matrix X + Mn. It is shown that there exist positive constants A and B depending on K, Q, ρ only such that [Formula: see text] As an application of this result we prove the elliptic law for this class of matrices with non-identically distributed correlated entries.

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Fluctuations of extreme eigenvalues of sparse Erdős–Rényi graphs

Probability Theory and Related Fields ◽

10.1007/s00440-021-01054-4 ◽

2021 ◽

Author(s):

Yukun He ◽

Antti Knowles

Keyword(s):

Random Matrices ◽

Adjacency Matrix ◽

Random Variable ◽

Total Degree ◽

Commun Math Phys ◽

Sparse Random Matrices ◽

Extreme Eigenvalues ◽

Optimal Scale ◽

Technical Achievement ◽

Math Phys

AbstractWe consider a class of sparse random matrices which includes the adjacency matrix of the Erdős–Rényi graph $${{\mathcal {G}}}(N,p)$$ G ( N , p ) . We show that if $$N^{\varepsilon } \leqslant Np \leqslant N^{1/3-\varepsilon }$$ N ε ⩽ N p ⩽ N 1 / 3 - ε then all nontrivial eigenvalues away from 0 have asymptotically Gaussian fluctuations. These fluctuations are governed by a single random variable, which has the interpretation of the total degree of the graph. This extends the result (Huang et al. in Ann Prob 48:916–962, 2020) on the fluctuations of the extreme eigenvalues from $$Np \geqslant N^{2/9 + \varepsilon }$$ N p ⩾ N 2 / 9 + ε down to the optimal scale $$Np \geqslant N^{\varepsilon }$$ N p ⩾ N ε . The main technical achievement of our proof is a rigidity bound of accuracy $$N^{-1/2-\varepsilon } (Np)^{-1/2}$$ N - 1 / 2 - ε ( N p ) - 1 / 2 for the extreme eigenvalues, which avoids the $$(Np)^{-1}$$ ( N p ) - 1 -expansions from Erdős et al. (Ann Prob 41:2279–2375, 2013), Huang et al. (2020) and Lee and Schnelli (Prob Theor Rel Fields 171:543–616, 2018). Our result is the last missing piece, added to Erdős et al. (Commun Math Phys 314:587–640, 2012), He (Bulk eigenvalue fluctuations of sparse random matrices. arXiv:1904.07140), Huang et al. (2020) and Lee and Schnelli (2018), of a complete description of the eigenvalue fluctuations of sparse random matrices for $$Np \geqslant N^{\varepsilon }$$ N p ⩾ N ε .

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