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Local circular law for random matrices. (English) Zbl 1301.15021
J. Ginibre [J. Math. Phys. 6, 440–449 (1965; Zbl 0127.39304)] had proved that the empirical spectral distribution of the eigenvalues of a complex matrix \(X\) of dimension \(N\times N\) with independent entries \(N^{-1/2} X_{ij}\), where \( X_{ij}\) are identically distributed according to the standard complex Gaussian measure, follows a circular law, i.e., converges to the uniform measure on the unit circle. In the case of real Gaussian entries, the limiting circular law was proved by A. Edelman [J. Multivariate Anal. 60, No. 2, 203–232 (1997; Zbl 0886.15024)].
For non-Gaussian entries, by using the Hermitization technique which allows the translation of the convergence of complex empirical measures into the convergence of logarithmic transforms for a family of Hermitian matrices [V. L. Girko, Teor. Veroyatn. Primen. 29, No. 4, 669–679 (1984; Zbl 0565.60034)], it is partially proved that the spectral measure of a non-Hermitian matrix \(X\) with independent entries converges to the circular law., i.e., if \(\mu_{j}\), \(j=1, 2,\dots, N\), are the eigenvalues of \(X\), then for any \(\mathcal{C}^2 \) function \(F\) \[ F \frac{1}{N} \sum_{j=1}^{N} F(\mu_{j}) = \frac{1}{4\pi N}\int \triangle F(z) \mathrm{Tr} \log (X^*-z^*)(X- z) dA(z).\tag{1} \] The aim of the contribution under review is to prove a local version of the circular law, up to the optimal scale \(N^{-1/2 + \varepsilon}\).
A key idea is the so-called stochastically domination. Let \(W= (W_{N})_{N\geq1}\) be a family of random variables and \(\Psi= (\Psi_{N})_{N\geq1}\) deterministic parameters. \(W\) is said to be stochastically dominated by \(\Psi\) (\(W \prec \Psi\) if for any positive real numbers \(\sigma\) and \(D\), and \(N\) large enough, one has \(\mathrm{P}[|W_{N}| > N^{\sigma} \Psi_{N}] \leq N^{-D}\).
Thus, the main result of the contribution (Theorem 2.2) reads as follows.
Let \(X= (X_{i,j})_{i,j=1}^{N}\) be a matrix with independent centered entries of variance \(N^{-1}\). Let assume that the probability distribution of the matrix entries satisfies a uniformly sub-exponentially decay condition \(\sup_{1\leq i,j\leq N} \operatorname P (|N^{1/2} X_{i,j} > \lambda)\leq \vartheta^{-1} \exp{- \lambda^\vartheta}\) for some positive real number \(\vartheta\) independent of \(N\). Assume that for some fixed \(\tau >0\) and for any \(N\) the inequality \(\tau \leq ||z_{0}|-1| \leq \tau^{-1}\) holds. Let \(f\) be a smooth function depending on \(N\) such that \(||f||_{\infty}\leq C\), \(||f'||_{\infty}\leq N^{C},\) and \(f(z) =0\) outside the disk of radius \(C\) (independent on \(N\)). Let \(f_{z_{0}}(z) = N^{2a} f (N^{a}(z-z_{0}))\) a scaling function around \(z_{0}\). Then, for any \(a\in (0, 1/2]\) the following stochastic domination property holds \[ \frac{1}{N} \sum_{j=1}^{N} f_{z_{0}} (\mu_{j})- \frac{1}{\pi}\int f_{z_{0}} d A(z) \prec N^{-1+ 2a} ||\triangle f||_{L^{1}}. \] The main tool is the analysis of the self-consistent equations satisfied by the Green functions \(G_{ij}(w)=[ (X^*- z^*)(X-z) - w]^{-1}_{i,j}\). The method is related to the proof of a local semicircular law or to local Marchenko-Pastur law. Indeed, there is a control of \(G_{ij}(E + i\eta)\) for the energy parameter in compact sets and small \(\eta\). In such a way the identity (1) plays a key role. Weak and strong local Green estimates are proved.

15B52 Random matrices (algebraic aspects)
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
60B20 Random matrices (probabilistic aspects)
15A18 Eigenvalues, singular values, and eigenvectors
Full Text: DOI arXiv
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