We show that the fluctuations of the largest eigenvalue of a real symmetric or complex Hermitian Wigner matrix of size N converge to the Tracy–Widom laws at a rate $$O(N^{-1/3+\omega })$$
O
(
N
-
1
/
3
+
ω
)
, as N tends to infinity. For Wigner matrices this improves the previous rate $$O(N^{-2/9+\omega })$$
O
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N
-
2
/
9
+
ω
)
obtained by Bourgade (J Eur Math Soc, 2021) for generalized Wigner matrices. Our result follows from a Green function comparison theorem, originally introduced by Erdős et al. (Adv Math 229(3):1435–1515, 2012) to prove edge universality, on a finer spectral parameter scale with improved error estimates. The proof relies on the continuous Green function flow induced by a matrix-valued Ornstein–Uhlenbeck process. Precise estimates on leading contributions from the third and fourth order moments of the matrix entries are obtained using iterative cumulant expansions and recursive comparisons for correlation functions, along with uniform convergence estimates for correlation kernels of the Gaussian invariant ensembles.
We consider N by N deformed Wigner random matrices of the form X N = H N +A N , where H N is a real symmetric or complex Hermitian Wigner matrix and A N is a deterministic real bounded diagonal matrix. We prove a universal Central Limit Theorem for the linear eigenvalue statistics of X N for all mesoscopic scales both in the spectral bulk and at regular edges where the global eigenvalue density vanishes as a square root. The method relies on the characteristic function method in [47], local laws for the Green function of X N in [3,46,51] and analytic subordination properties of the free additive convolution [24,41]. We also prove the analogous results for high-dimensional sample covariance matrices.
We consider random matrices of the form $H_N=A_N+U_N B_N U^*_N$, where $A_N$ and $B_N$ are two $N$ by $N$ deterministic Hermitian matrices and $U_N$ is a Haar distributed random unitary matrix. We establish a universal central limit theorem for the linear eigenvalue statistics of $H_N$ on all mesoscopic scales inside the regular bulk of the spectrum. The proof is based on studying the characteristic function of the linear eigenvalue statistics and consists of two main steps: (1) generating Ward identities using the left-translation invariance of the Haar measure, along with a local law for the resolvent of $H_N$ and analytic subordination properties of the free additive convolution, allows us to derive an explicit formula for the derivative of the characteristic function; (2) a local law for two-point product functions of resolvents is derived using a partial randomness decomposition of the Haar measure. We also prove the corresponding results for orthogonal conjugations.
We consider the eigenvalues of a large dimensional real or complex Ginibre matrix in the region of the complex plane where their real parts reach their maximum value. This maximum follows the Gumbel distribution and that these extreme eigenvalues form a Poisson point process as the dimension asymptotically tends to infinity. In the complex case, these facts have already been established by Bender [Probab. Theory Relat. Fields 147, 241 (2010)] and in the real case by Akemann and Phillips [J. Stat. Phys. 155, 421 (2014)] even for the more general elliptic ensemble with a sophisticated saddle point analysis. The purpose of this article is to give a very short direct proof in the Ginibre case with an effective error term. Moreover, our estimates on the correlation kernel in this regime serve as a key input for accurately locating [Formula: see text] for any large matrix X with i.i.d. entries in the companion paper [G. Cipolloni et al., arXiv:2206.04448 (2022)].
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