Fluctuations of matrix entries of analytic functions of non-Hermitian random matrices (Q2920379)

Consider an \(n\times n\) non-Hermitian random matrix \(M_n=(m_{nij})_{i,j=1}^n\) whose entries are independent real random variables. Under a fourth moment condition on the entries, the author studies the fluctuations of the entries of the matrix \(f(M_n/\sqrt n)\) as \(n\to\infty\), where \(f\) is an analytic function on the disk \(\{z\in\mathbb C: |z|<2+\varepsilon\}\). In the main result, the author shows that, for every fixed \(i,j\in\mathbb N\), the normalized entry NEWLINE\[NEWLINE \sqrt n\left(f\left(\frac {M_n} {\sqrt n}\right)_{ij}- f(0)\delta_{ij} - \frac{f'(0)}{\sqrt n} m_{nij}\right) NEWLINE\]NEWLINE converges in distribution to a complex Gaussian random variable, as \(n\to\infty\). Here, \(\delta_{ij}\) denotes the Kronecker delta symbol. He also shows that different entries become asymptotically independent. As a special case of this setting, the author studies the entries of the resolvent NEWLINE\[NEWLINE \left(z \cdot \text{Id}_n- \frac{M_n}{\sqrt n}\right)^{-1}. NEWLINE\]NEWLINE He shows that the entries of the resolvent, considered as random functions of \(z\in \mathbb C\), \(|z|>2+\varepsilon\), converge weakly on the space of continuous functions to a certain Gaussian process, as \(n\to\infty\). The paper extends a number of previous results from symmetric random matrices to the non-symmetric setting.