The circular law for random matrices

By Friedrich Götze, A. Tikhomirov. 2007

179 citations recorded by the source corpus.

We consider the joint distribution of real and imaginary parts of eigenvalues of random matrices with independent real entries with mean zero and unit variance. We prove the convergence of this distribution to the uniform distribution on the unit disc without assumptions on the existence of a density for the distribution of entries. We assume that the entries have a finite moment of order larger than two and consider the case of sparse matrices. The results are based on previous work of Bai, Rudelson and the authors extending results to a larger class of sparse matrices.

Subfields: Management Science and Operations Research, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/18aec0fe25451988ef1162852cf614cab37fa545

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In Management Science and Operations Research

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Universality and the circular law for sparse random matrices. (2010, background): “The strongest result in the literature for nonsymmetric sparse random matrices is due to Tao and Vu [32] who in 2008 proved a sparse version of the circular law with the assumption of bounded (2 + δ)th moment (note that [32] proves almost sure convergence, rather than convergence in probability as s”
The single ring theorem (2009, background): “It gives an implicit equation forG(·) in terms ofGT(·).”
The Elliptic Law (2012, methodology): “ general (non-Gaussian) case when there is no formula, the problem appears much more dicult. Important results were obtained by Girko [12, 13], Bai [1, 3], and more recently by Gotze and Tikhomirov [19], Pan and Zhou [36], and Tao and Vu [45]. These results conrm the same limiting law under some mom”
Non-asymptotic theory of random matrices: extreme singular values (2010, background): “This was further extended to random variables having bounded moment 2 + ε in [35, 82].”
Small Ball Probability, Inverse Theorems, and Applications (2012, background): “For applications in Random Matrix Theory (such as the establishment of the Circular Law) and many related results, we refer to [59, 65, 58, 18, 40, 2, 47] and the references therein.”
Spectrum of Non-Hermitian Heavy Tailed Random Matrices (2010, methodology): “The removal of this hypothesis can be done by adapting the Rudelson and Vershynin approach already used by Götze and Tikhomirov [28].”
Invertibility of Sparse non-Hermitian matrices (2015, background): “In [10], under a minimal moment assumption, it was also shown that smax(An) ≤ n √ pn with probability tending to 1.”
On the Asymptotic Spectrum of Products of Independent Random Matrices. (2010, methodology): “For m = 1 we recover the well-known circular law for random matrices [9], [15].”
Around the circular law (2011, background): “The terminology corresponds to what we actually draw: a circle for the circular law, a quarter circle (actually a quarter ellipse) for the quarter circular law, even if it is the boundary of the support in the first case, and the density in the second case.”
From the Littlewood-Offord problem to the Circular Law: universality of the spectral distribution of random matrices (2008, background): “An important issue in the theory of computing is noise, as almost all computational processes are effected by it.”
On the Marchenko-Pastur and Circular Laws for some Classes of Random Matrices with Dependent Entries (2011, background): “[16, 17, 8, 7, 28, 19, 36]) and to our best knowledge, the only article concerning nonindependent entries is [9] examining the case of random Markov matrices.”
Outliers in the spectrum of iid matrices with bounded rank perturbations (2010, background): “ C !C, R C F d A n converges in probability (resp. almost surely) to R C Fd. The following theorem is the culmination of the work of many authors [23], [33], [24], [4], [6], [26], [25], [34], [42], [27], [44]: Theorem 1.3 (Circular law for iid matrices). Let X n be an iid random matrix. Then p1 n X”
Spectrum of Markov Generators on Sparse Random Graphs (2012, methodology): “10) improves over previous works [23, 45], where the variance is assumed to satisfy n1−εσ2 → ∞ for some ε > 0.”
Random matrices: Universality of ESDs and the circular law (2008, result): “mplex random variables with mean 0 and variance 1. Then the ESD of p1 n X n converges (in both the strong and weak senses) to the uniform distribution on the unit disk. Remark 1.16. In [26] (see also [10] for an alternate proof for the weak version), this theorem was proven with the extra assumption”