Circular law for random matrices with exchangeable entries

Adamczak, Radosław; Chafäı, Djalil; Wolff, Paweł

doi:10.1002/rsa.20599

Cited by 33 publications

(50 citation statements)

References 32 publications

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“…. , x σ(n) ), we recover the deviation inequality by Adamczak, Chafaï and Wolff [ACW14] (Theorem 3.1) obtained from Theorem 5.6 by Talagrand. This concentration property plays a key role in their approach, to study the convergence of the empirical spectral measure of random matrices with exchangeable entries, when the size of these matrices is increasing.…”

supporting

confidence: 75%

Concentration of Measure Principle and Entropy-Inequalities

Samson

2017

Convexity and Concentration

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Abstract. The concentration measure principle is presented in an abstract way to encompass and unify different concentration properties. We give a general overview of the links between concentration properties, transport-entropy inequalities, and logarithmic Sobolev inequalities for some specific transport costs. By giving few examples, we emphasize optimal weak transport costs as an efficient tool to establish new transport inequality and new concentration principles for discrete measures (the binomial law, the Poisson measure, the uniform law on the symmetric group).

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supporting

confidence: 75%

Concentration of Measure Principle and Entropy-Inequalities

Samson

2017

Convexity and Concentration

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“…(1) n = 1 √ npn A n and B (2) n which is the matrix of i.i.d. centered complex Gaussian entries with variance 1/n.…”

Section: Preliminaries and Proof Outlinementioning

confidence: 99%

The circular law for sparse non-Hermitian matrices

Basak¹,

Rudelson²

2019

Ann. Probab.

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For a class of sparse random matrices of the form An = (ξ i,j δ i,j ) n i,j=1 , where {ξ i,j } are i.i.d. centered sub-Gaussian random variables of unit variance, and {δ i,j } are i.i.d. Bernoulli random variables taking value 1 with probability pn, we prove that the empirical spectral distribution of An/ √ npn converges weakly to the circular law, in probability, for all pn such that pn = ω(log 2 n/n). Additionally if pn satisfies the inequality npn > exp(c √ log n) for some constant c, then the above convergence is shown to hold almost surely. The key to this is a new bound on the smallest singular value of complex shifts of real valued sparse random matrices. The circular law limit also extends to the adjacency matrix of a directed Erdős-Rényi graph with edge connectivity probability pn.MSC 2010 subject classifications: 15B52, 60B10, 60B20.

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“…to obtain (26). To do so, the idea, as in [ACW14], is to show that the assertion f (A, Π n ) < t 2 implies that √ Z < √ C A + t max 1≤i,j≤n {a i,j }, and to conclude by contraposition. Then, the two following steps consist in choosing appropriate constants C A in (26) depending on the median of Z, such that both P √ Z ≥ √ C A + t max 1≤i,j≤n {a i,j } and P(Z ∈ A) are greater than 1/2, in order to control both probabilities…”

Section: Proof Of Lemma 21mentioning

confidence: 99%

Concentration Inequalities for Randomly Permuted Sums

Albert

2019

Progress in Probability

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Initially motivated by the study of the non-asymptotic properties of non-parametric tests based on permutation methods, concentration inequalities for uniformly permuted sums have been largely studied in the literature. Recently, Delyon et al. proved a new Bernstein-type concentration inequality based on martingale theory. This work presents a new proof of this inequality based on the fundamental inequalities for random permutations of Talagrand. The idea is to first obtain a rough inequality for the square root of the permuted sum, and then, iterate the previous analysis and plug this first inequality to obtain a general concentration of permuted sums around their median. Then, concentration inequalities around the mean are deduced. This method allows us to obtain the Bernstein-type inequality up to constants, and, in particular, to recovers the Gaussian behavior of such permuted sums under classical conditions encountered in the literature. Then, an application to the study of the second kind error rate of permutation tests of independence is presented.Mathematics Subject Classification: 60E15, 60C05.

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Circular law for random matrices with exchangeable entries

Cited by 33 publications

References 32 publications

Concentration of Measure Principle and Entropy-Inequalities

Concentration of Measure Principle and Entropy-Inequalities

The circular law for sparse non-Hermitian matrices

Concentration Inequalities for Randomly Permuted Sums

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