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Random matrices: universality of local spectral statistics of non-Hermitian matrices. (English) Zbl 1316.15042
The complex Gaussian matrix ensemble is constituted by random matrices whose entries are independent and identically distributed with the distribution of a complex Gaussian \(N(0, 1/2)_{\mathbf{C}}\) with mean zero and variance one, i.e., the probability distribution of each entry is \(\omega(z)= \frac{1}{\pi} \exp (- |z|^{2}),\) and the real and imaginary parts of each entry independently have the distribution \(N(0, 1/2)_{\mathbf{R}}.\) The correlation functions \(\rho_{n}^{(k)}(z_{1}, \cdot\cdot\cdot, z_{k})\) of a complex Gaussian matrix are given by the Ginibre formula (see [J. Ginibre, J. Math. Phys. 6, 440–449 (1965; Zbl 0127.39304)]) \(\rho_{n}^{(k)}(z_{1}, \cdot\cdot\cdot, z_{k})= \det (K_{n}(z_{i}, z_{j}))_{i,j=1}^{k}\), where \(K_{n}(z,y)= \omega(z)^{1/2} \omega(y)^{1/2} \sum_{j=0}^{n-1} \frac{(z\bar{y})^{j}}{j!}\) is the \(n\)th kernel associated with the Gaussian distribution. For \(k=1\), one has the asymptotics \(\rho_{n}^{(1)} (n^{1/2} z) \rightarrow \frac{1}{\pi} 1_{|z|\leq 1}\) a.e. which yields the circular law for complex Gaussian matrices.
In the paper under review, the authors prove, as a first result, the universality of the asymptotic law for Gaussian ensembles among all random \(n\times n\) matrices whose entries are jointly independent, have independent real and imaginary parts, exponentially decaying, the so-called (C1) condition, and whose moments match those of the complex Gaussian ensemble to fourth order.
In the Hermitian case, four moment theorems can be used to extend various facts concerning the asymptotic spectral distribution of special matrix ensembles to other matrix ensembles which satisfy appropriate moment matching conditions. In such a sense, a partial extension of a central limit theorem (see [B. Rider, Probab. Theory Relat. Fields 130, No. 3, 337–367 (2004; Zbl 1071.82029)]) in the small radius case is deduced. In the case of real matrices, a four moment theorem, an universality result as well as the asymptotic behavior of real matrices satisfying the (C1) condition are also obtained. As a direct and quick application, the authors show that for many ensembles of independent-entry matrices \(M_{n}\) satisfying the condition (C1) and matching moments with the real or complex Gaussian matrix to fourth order (in the real case it is assumed that \(n\) is even) most of the eigenvalues are simple in the sense that with probability \(1-O(n^{-c})\), at most \(O(n^{1-c})\) of the complex eigenvalues, and \(O(n^{1/2-c})\) of the real eigenvalues, are repeated, for some fixed \(c>0\).
Since the spectrum of non-Hermitian matrices is unstable, the authors use the log-determinants \(\log|\det (M_{n}- z_{0})|\) instead of the resolvent \((M_{n}-z_{0})^{-1}\) or the closely related Stieltjes transform \(\frac{1}{n} \operatorname{trace}(M_{n}-z_{0})\) taking into account the connection between spectral statistics and the log-determinant which goes back to the Girko’s Hermitization method (see [V. L. Girko, Teor. Veroyatn. Primen. 29, No. 4, 669–679 (1984; Zbl 0565.60034)]). Notice that the log-determinant is connected to the eigenvalues of the independent and identically distributed matrix \(M_{n}\) via the identity \(\log|\det (M_{n}- z_{0})|= \sum_{i=0}^{n} \log| \lambda_{i}(M_{n})- z_{0}|.\) The main tools are a four moment theorem for these log-determinants together with a strong concentration result for the log-determinants in the Gausssian case. This is done from the analysis of the solutions of a certain nonlinear stochastic difference equation which is governed by the dynamics of the maps \(a\mapsto \frac{|z_{0}|a n^{1/2}}{(|a|^{2} + n -i)^{1/2}}\) as \(i\) increases from \(1\) to \(n-1\). A crude lower bound, lower bounds at early times, concentration at late times based on repulsive properties near the origin to propagate the initial largeness to latter values of \(i\) are obtained for real and Gaussian matrices. From here, the concentration bound on log-determinant for an independent-entry matrix \(M_{n}\) satisfying the (C1) condition and matching the real or complex Gaussian ensembles to third order means that for any fixed \(C>0\) and any \(z_{0}\in B(0, C)\), \(\log|\det (M_{n}- z_{0} n^{1/2})|\) concentrates around \(\frac{n \log n}{2} + \frac{1}{2} n (|z_{0}|^{2}-1)\) for \(|z_{0}|\leq 1\) and around \(\frac{n \log n}{2} + n \log |z_{0}|\) for \(|z_{0}|\geq 1\), uniformly in \(|z_{0}|\).

MSC:
15B52 Random matrices (algebraic aspects)
60B20 Random matrices (probabilistic aspects)
15A15 Determinants, permanents, traces, other special matrix functions
39A50 Stochastic difference equations
44A15 Special integral transforms (Legendre, Hilbert, etc.)
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