Around the circular law

By Charles Bordenave, Djalil CHAFAÏ. 2011

243 citations recorded by the source corpus.

These expository notes are centered around the circular law theorem, which states that the empirical spectral distribution of a nxn random matrix with i.i.d. entries of variance 1/n tends to the uniform law on the unit disc of the complex plane as the dimension $n$ tends to infinity. This phenomenon is the non-Hermitian counterpart of the semi circular limit for Wigner random Hermitian matrices, and the quarter circular limit for Marchenko-Pastur random covariance matrices. We present a proof in a Gaussian case, due to Silverstein, based on a formula by Ginibre, and a proof of the universal case by revisiting the approach of Tao and Vu, based on the Hermitization of Girko, the logarithmic potential, and the control of the small singular values. Beyond the finite variance model, we also consider the case where the entries have heavy tails, by using the objective method of Aldous and Steele borrowed from randomized combinatorial optimization. The limiting law is then no longer the circular law and is related to the Poisson weighted infinite tree. We provide a weak control of the smallest singular value under weak assumptions, using asymptotic geometric analysis tools. We also develop a quaternionic Cauchy-Stieltjes transform borrowed from the Physics literature.

Subfields: Computational Theory and Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/d6a19dd9a4404cf5e904d53e235577c555dd0b91

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

The Elliptic Law (2012, methodology): “atrices such that rank(F n) = o(n) and sup n 1 n2 kF k 2 2 <1. Then for a.a. z 2C a.s. log is uniformly integrable for f p1 n (Xn+Fn) zI g n1. The proof of Lemma 7.5 is based on the arguments of [4, 29, 42]. In order to prove Lemma 7.5, we will need the following bound for small singular values. ”
Lower bounds for the smallest singular value of structured random matrices (2016, background): “1 in [6]) relaxes the hypothesis of continuous distributions from Proposition 1.”
Spectrum of Markov Generators on Sparse Random Graphs (2012, methodology): “, to [11,13] for the spectral analysis of non-Hermitian matrices with i.”
Asymptotic distribution of complex zeros of random analytic functions (2014, background): “ distribution is uniform on D; see Figure 1. This result can be seen as an analogue of the famous circular law for the distribution of eigenvalues of non-Hermitian random matrices with i.i.d. entries [5, 35]. Forrester and Honner [9] stated the circular law for Weyl polynomials and discussed diﬀeren”
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.”
Cleaning large correlation matrices: tools from random matrix theory (2016, background): “Dirac delta function. Note that the symmetry of the considered matrices ensures that the eigenvalues of M are dened on the real line (complex eigenvalues are beyond the scope of this review, but see [55,56,61] for more on this). One of the most important property of large random matrices is that one”
Random matrices: Universality of local spectral statistics of non-Hermitian matrices (2012, methodology): “There is a non-trivial issue with the possible divergence or instability of the integral in (18) near η = 0, but it is now well understood how to control this issue via a regularisation or truncation of this integral, provided that one has adequate bounds on the least singular value of Wn,z; again, ”
Outlier Eigenvalues for Deformed I.I.D. Random Matrices (2014, methodology): “For more references, we refer to the surveys [17, 54].”
On the singularity of adjacency matrices for random regular digraphs (2014, background): “It was conjectured in [6] that the same law should hold for the sum of a fixed number of independent uniform random permutation matrices, which can be viewed as a sparse version of the rrd matrix M .”
Random Polynomials and Pluripotential-Theoretic Extremal Functions (2013, background): “ft-hand inequality follows from the deﬁnition of φ(n) Y,Q(z) in (6.4). Theorems 4.1, 4.2, and 5.1 in this setting follow immediately. As an application, we consider random Weyl polynomials in C (cf., [2]). These are random polynomials in one variable of the form Xn j=0 anj zj √ j! where the anj are ”
Invertibility of Sparse non-Hermitian matrices (2015, unknown): “It is well known that a key step to such analysis is the control on smin(Ān − ω √ npIn) for ω ∈ B(0, R) ⊂ C, for some R finite (see [7]).”