Sparse random matrices have simple spectrum

Luh, Kyle; Vu, Van

doi:10.1214/19-aihp1032

Cited by 6 publications

(23 citation statements)

References 22 publications

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“…Let λ 1 ≥ • • • ≥ λ N be the ordered eigenvalues of H. We consider an orthonormal basis of eigenvectors of H by {v 1 , • • • , v N }, i.e., Hv i = λ i v i and v i = 1 for each i. Note that Luh and Vu recently showed that sparse random matrices have simple spectrum [24] with probability tending to one as N goes to infinity. This implies that λ 1 > • • • > λ N and the eigenvectors are uniquely determined up to a sign.…”

Section: Definition and Main Resultsmentioning

confidence: 99%

Noise sensitivity of the top eigenvector of a Wigner matrix

Bordenave

Lugosi

Zhivotovskiy

2020

Probab. Theory Relat. Fields

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We investigate the noise sensitivity of the top eigenvector of a Wigner matrix in the following sense. Let v be the top eigenvector of an N × N Wigner matrix. Suppose that k randomly chosen entries of the matrix are resampled, resulting in another realization of the Wigner matrix with top eigenvector v [k] . We prove that, with high probability, when k ≪ N 5/3−o(1) , then v and v [k] are almost collinear and when k ≫ N 5/3 , then v [k] is almost orthogonal to v.

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Section: Definition and Main Resultsmentioning

confidence: 99%

Noise sensitivity of the top eigenvector of a Wigner matrix

Bordenave

Lugosi

Zhivotovskiy

2020

Probab. Theory Relat. Fields

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“…Observe that |V α | ∞ = 1 and hence D (V α ) ≥ 1/2. Therefore, inequality (K), inequality (15) and inequality (16) yield…”

Section: Estimation On J 1 (N N)mentioning

confidence: 89%

“…We remark that the lcd has been converted into a useful tool that allows the analysis of various interesting problems. For instance, it is used in the study of isomorphism between graphs [15] and in the analysis of the condition number for random matrices [18]. In this paper, the lcd is used to understand how small the modulus of a random polynomial near the unit circle can be.…”

Section: Introductionmentioning

confidence: 99%

Zero-free neighborhoods around the unit circle for Kac polynomials

Barrera

Manrique

2021

Period Math Hung

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In this paper, we study how the roots of the Kac polynomials $$W_n(z) = \sum _{k=0}^{n-1} \xi _k z^k$$ W n ( z ) = ∑ k = 0 n - 1 ξ k z k concentrate around the unit circle when the coefficients of $$W_n$$ W n are independent and identically distributed nondegenerate real random variables. It is well known that the roots of a Kac polynomial concentrate around the unit circle as $$n\rightarrow \infty $$ n → ∞ if and only if $${\mathbb {E}}[\log ( 1+ |\xi _0|)]<\infty $$ E [ log ( 1 + | ξ 0 | ) ] < ∞ . Under the condition $${\mathbb {E}}[\xi ^2_0]<\infty $$ E [ ξ 0 2 ] < ∞ , we show that there exists an annulus of width $${\text {O}}(n^{-2}(\log n)^{-3})$$ O ( n - 2 ( log n ) - 3 ) around the unit circle which is free of roots with probability $$1-{\text {O}}({(\log n)^{-{1}/{2}}})$$ 1 - O ( ( log n ) - 1 / 2 ) . The proof relies on small ball probability inequalities and the least common denominator used in [17].

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“…However, due to the low rank structure of pJ, we can extend the covering arguments to handle this case (cf. [7,57,53]). We let E K denote the event that…”

Section: Directed Erdős-rényi Random Graphsmentioning

confidence: 99%

Eigenvectors and controllability of non-Hermitian random matrices and directed graphs

Luh

O’Rourke

2021

Electron. J. Probab.

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We study the eigenvectors and eigenvalues of random matrices with iid entries. Let N be a random matrix with iid entries which have symmetric distribution. For each unit eigenvector v of N our main results provide a small ball probability bound for linear combinations of the coordinates of v. Our results generalize the works of Meehan and Nguyen [59] as well as Touri and the second author [67,68,69] for random symmetric matrices. Along the way, we provide an optimal estimate of the probability that an iid matrix has simple spectrum, improving a recent result of Ge [37]. Our techniques also allow us to establish analogous results for the adjacency matrix of a random directed graph, and as an application we establish controllability properties of network control systems on directed graphs.

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Sparse random matrices have simple spectrum

Cited by 6 publications

References 22 publications

Noise sensitivity of the top eigenvector of a Wigner matrix

Noise sensitivity of the top eigenvector of a Wigner matrix

Zero-free neighborhoods around the unit circle for Kac polynomials

Eigenvectors and controllability of non-Hermitian random matrices and directed graphs

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