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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.

MSC:
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
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