@article{20328, abstract = {We consider the standard overlap (math formular) of any bi-orthogonal family of left and right eigenvectors of a large random matrix X with centred i.i.d. entries and we prove that it decays as an inverse second power of the distance between the corresponding eigenvalues. This extends similar results for the complex Gaussian ensemble from Bourgade and Dubach [15], as well as Benaych-Georges and Zeitouni [13], to any i.i.d. matrix ensemble in both symmetry classes. As a main tool, we prove a two-resolvent local law for the Hermitisation of X uniformly in the spectrum with optimal decay rate and optimal dependence on the density near the spectral edge.}, author = {Cipolloni, Giorgio and Erdös, László and Xu, Yuanyuan}, issn = {0022-1236}, journal = {Journal of Functional Analysis}, number = {1}, publisher = {Elsevier}, title = {{Optimal decay of eigenvector overlap for non-Hermitian random matrices}}, doi = {10.1016/j.jfa.2025.111180}, volume = {290}, year = {2026}, } @article{21271, abstract = {For general non-Hermitian large random matrices X and deterministic deformation matrices A, we prove that the local eigenvalue statistics of A+X close to the typical edge points of its spectrum are universal. Furthermore, we show that, under natural assumptions, on A the spectrum of A+X does not have outliers at a distance larger than the natural fluctuation scale of the eigenvalues. As a consequence, the number of eigenvalues in each component of Spec(A+X) is deterministic.}, author = {Campbell, Andrew J and Cipolloni, Giorgio and Erdös, László and Ji, Hong Chang}, issn = {2168-894X}, journal = {The Annals of Probability}, number = {6}, pages = {2256--2308}, publisher = {Institute of Mathematical Statistics}, title = {{On the spectral edge of non-Hermitian random matrices}}, doi = {10.1214/25-aop1761}, volume = {53}, year = {2025}, } @unpublished{19546, abstract = {We study the sensitivity of the eigenvectors of random matrices, showing that even small perturbations make the eigenvectors almost orthogonal. More precisely, we consider two deformed Wigner matrices $W+D_1$, $W+D_2$ and show that their bulk eigenvectors become asymptotically orthogonal as soon as $\mathrm{Tr}(D_1-D_2)^2\gg 1$, or their respective energies are separated on a scale much bigger than the local eigenvalue spacing. Furthermore, we show that quadratic forms of eigenvectors of $W+D_1$, $W+D_2$ with any deterministic matrix $A\in\mathbf{C}^{N\times N}$ in a specific subspace of codimension one are of size $N^{-1/2}$. This proves a generalization of the Eigenstate Thermalization Hypothesis to eigenvectors belonging to two different spectral families.}, author = {Cipolloni, Giorgio and Erdös, László and Henheik, Sven Joscha and Kolupaiev, Oleksii}, booktitle = {arXiv}, title = {{Eigenvector decorrelation for random matrices}}, doi = {10.48550/arXiv.2410.10718}, year = {2025}, } @article{19737, abstract = {For general large non–Hermitian random matrices X and deterministic normal deformations A, we prove that the local eigenvalue statistics of A + X close to the critical edge points of its spectrum are universal. This concludes the proof of the third and last remaining typical universality class for non–Hermitian random matrices (for normal deformations), after bulk and sharp edge universalities have been established in recent years.}, author = {Cipolloni, Giorgio and Erdös, László and Ji, Hong Chang}, issn = {1432-2064}, journal = {Probability Theory and Related Fields}, publisher = {Springer Nature}, title = {{Non–Hermitian spectral universality at critical points}}, doi = {10.1007/s00440-025-01384-7}, year = {2025}, } @article{20478, abstract = {We consider the Wigner minor process, i.e. the eigenvalues of an N\times N Wigner matrix H^{(N)} together with the eigenvalues of all its n\times n minors, H^{(n)}, n\le N. The top eigenvalues of H^{(N)} and those of its immediate minor H^{(N-1)} are very strongly correlated, but this correlation becomes weaker for smaller minors H^{(N-k)} as k increases. For the GUE minor process the critical transition regime around k\sim N^{2/3} was analyzed by Forrester and Nagao (J. Stat. Mech.: Theory and Experiment, 2011) providing an explicit formula for the nontrivial joint correlation function. We prove that this formula is universal, i.e. it holds for the Wigner minor process. Moreover, we give a complete analysis of the sub- and supercritical regimes both for eigenvalues and for the corresponding eigenvector overlaps, thus we prove the decorrelation transition in full generality.}, author = {Bao, Zhigang and Cipolloni, Giorgio and Erdös, László and Henheik, Sven Joscha and Kolupaiev, Oleksii}, issn = {1432-2064}, journal = {Probability Theory and Related Fields}, publisher = {Springer Nature}, title = {{Decorrelation transition in the Wigner minor process}}, doi = {10.1007/s00440-025-01422-4}, year = {2025}, } @article{14408, abstract = {We prove that the mesoscopic linear statistics ∑if(na(σi−z0)) of the eigenvalues {σi}i of large n×n non-Hermitian random matrices with complex centred i.i.d. entries are asymptotically Gaussian for any H20-functions f around any point z0 in the bulk of the spectrum on any mesoscopic scale 0