@article{11741,
  abstract     = {Following E. Wigner’s original vision, we prove that sampling the eigenvalue gaps within the bulk spectrum of a fixed (deformed) Wigner matrix H yields the celebrated Wigner-Dyson-Mehta universal statistics with high probability. Similarly, we prove universality for a monoparametric family of deformed Wigner matrices H+xA with a deterministic Hermitian matrix A and a fixed Wigner matrix H, just using the randomness of a single scalar real random variable x. Both results constitute quenched versions of bulk universality that has so far only been proven in annealed sense with respect to the probability space of the matrix ensemble.},
  author       = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
  issn         = {1432-2064},
  journal      = {Probability Theory and Related Fields},
  pages        = {1183–1218},
  publisher    = {Springer Nature},
  title        = {{Quenched universality for deformed Wigner matrices}},
  doi          = {10.1007/s00440-022-01156-7},
  volume       = {185},
  year         = {2023},
}

@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<a<1/2. This extends our previous result (Cipolloni et al. in Commun Pure Appl Math, 2019. arXiv:1912.04100), that was valid on the macroscopic scale, a=0
, to cover the entire mesoscopic regime. The main novelty is a local law for the product of resolvents for the Hermitization of X at spectral parameters z1,z2 with an improved error term in the entire mesoscopic regime |z1−z2|≫n−1/2. The proof is dynamical; it relies on a recursive tandem of the characteristic flow method and the Green function comparison idea combined with a separation of the unstable mode of the underlying stability operator.},
  author       = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
  issn         = {1432-2064},
  journal      = {Probability Theory and Related Fields},
  publisher    = {Springer Nature},
  title        = {{Mesoscopic central limit theorem for non-Hermitian random matrices}},
  doi          = {10.1007/s00440-023-01229-1},
  year         = {2023},
}

@article{14849,
  abstract     = {We establish a precise three-term asymptotic expansion, with an optimal estimate of the error term, for the rightmost eigenvalue of an n×n random matrix with independent identically distributed complex entries as n tends to infinity. All terms in the expansion are universal.},
  author       = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J and Xu, Yuanyuan},
  issn         = {0091-1798},
  journal      = {The Annals of Probability},
  keywords     = {Statistics, Probability and Uncertainty, Statistics and Probability},
  number       = {6},
  pages        = {2192--2242},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{On the rightmost eigenvalue of non-Hermitian random matrices}},
  doi          = {10.1214/23-aop1643},
  volume       = {51},
  year         = {2023},
}

@article{10405,
  abstract     = {We consider large non-Hermitian random matrices X with complex, independent, identically distributed centred entries and show that the linear statistics of their eigenvalues are asymptotically Gaussian for test functions having 2+ϵ derivatives. Previously this result was known only for a few special cases; either the test functions were required to be analytic [72], or the distribution of the matrix elements needed to be Gaussian [73], or at least match the Gaussian up to the first four moments [82, 56]. We find the exact dependence of the limiting variance on the fourth cumulant that was not known before. The proof relies on two novel ingredients: (i) a local law for a product of two resolvents of the Hermitisation of X with different spectral parameters and (ii) a coupling of several weakly dependent Dyson Brownian motions. These methods are also the key inputs for our analogous results on the linear eigenvalue statistics of real matrices X that are presented in the companion paper [32]. },
  author       = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
  issn         = {1097-0312},
  journal      = {Communications on Pure and Applied Mathematics},
  number       = {5},
  pages        = {946--1034},
  publisher    = {Wiley},
  title        = {{Central limit theorem for linear eigenvalue statistics of non-Hermitian random matrices}},
  doi          = {10.1002/cpa.22028},
  volume       = {76},
  year         = {2023},
}

@article{12761,
  abstract     = {We consider the fluctuations of regular functions f of a Wigner matrix W viewed as an entire matrix f (W). Going beyond the well-studied tracial mode, Trf (W), which is equivalent to the customary linear statistics of eigenvalues, we show that Trf (W)A is asymptotically normal for any nontrivial bounded deterministic matrix A. We identify three different and asymptotically independent modes of this fluctuation, corresponding to the tracial part, the traceless diagonal part and the off-diagonal part of f (W) in the entire mesoscopic regime, where we find that the off-diagonal modes fluctuate on a much smaller scale than the tracial mode. As a main motivation to study CLT in such generality on small mesoscopic scales, we determine
the fluctuations in the eigenstate thermalization hypothesis (Phys. Rev. A 43 (1991) 2046–2049), that is, prove that the eigenfunction overlaps with any deterministic matrix are asymptotically Gaussian after a small spectral averaging. Finally, in the macroscopic regime our result also generalizes (Zh. Mat. Fiz. Anal. Geom. 9 (2013) 536–581, 611, 615) to complex W and to all crossover ensembles in between. The main technical inputs are the recent
multiresolvent local laws with traceless deterministic matrices from the companion paper (Comm. Math. Phys. 388 (2021) 1005–1048).},
  author       = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
  issn         = {1050-5164},
  journal      = {Annals of Applied Probability},
  number       = {1},
  pages        = {447--489},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{Functional central limit theorems for Wigner matrices}},
  doi          = {10.1214/22-AAP1820},
  volume       = {33},
  year         = {2023},
}

@article{12792,
  abstract     = {In the physics literature the spectral form factor (SFF), the squared Fourier transform of the empirical eigenvalue density, is the most common tool to test universality for disordered quantum systems, yet previous mathematical results have been restricted only to two exactly solvable models (Forrester in J Stat Phys 183:33, 2021. https://doi.org/10.1007/s10955-021-02767-5, Commun Math Phys 387:215–235, 2021. https://doi.org/10.1007/s00220-021-04193-w). We rigorously prove the physics prediction on SFF up to an intermediate time scale for a large class of random matrices using a robust method, the multi-resolvent local laws. Beyond Wigner matrices we also consider the monoparametric ensemble and prove that universality of SFF can already be triggered by a single random parameter, supplementing the recently proven Wigner–Dyson universality (Cipolloni et al. in Probab Theory Relat Fields, 2021. https://doi.org/10.1007/s00440-022-01156-7) to larger spectral scales. Remarkably, extensive numerics indicates that our formulas correctly predict the SFF in the entire slope-dip-ramp regime, as customarily called in physics.},
  author       = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
  issn         = {1432-0916},
  journal      = {Communications in Mathematical Physics},
  pages        = {1665--1700},
  publisher    = {Springer Nature},
  title        = {{On the spectral form factor for random matrices}},
  doi          = {10.1007/s00220-023-04692-y},
  volume       = {401},
  year         = {2023},
}

@article{10732,
  abstract     = {We compute the deterministic approximation of products of Sobolev functions of large Wigner matrices W and provide an optimal error bound on their fluctuation with very high probability. This generalizes Voiculescu's seminal theorem from polynomials to general Sobolev functions, as well as from tracial quantities to individual matrix elements. Applying the result to eitW for large t, we obtain a precise decay rate for the overlaps of several deterministic matrices with temporally well separated Heisenberg time evolutions; thus we demonstrate the thermalisation effect of the unitary group generated by Wigner matrices.},
  author       = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
  issn         = {1096-0783},
  journal      = {Journal of Functional Analysis},
  number       = {8},
  publisher    = {Elsevier},
  title        = {{Thermalisation for Wigner matrices}},
  doi          = {10.1016/j.jfa.2022.109394},
  volume       = {282},
  year         = {2022},
}

@article{11418,
  abstract     = {We consider the quadratic form of a general high-rank deterministic matrix on the eigenvectors of an N×N
Wigner matrix and prove that it has Gaussian fluctuation for each bulk eigenvector in the large N limit. The proof is a combination of the energy method for the Dyson Brownian motion inspired by Marcinek and Yau (2021) and our recent multiresolvent local laws (Comm. Math. Phys. 388 (2021) 1005–1048).},
  author       = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
  issn         = {2168-894X},
  journal      = {Annals of Probability},
  number       = {3},
  pages        = {984--1012},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{Normal fluctuation in quantum ergodicity for Wigner matrices}},
  doi          = {10.1214/21-AOP1552},
  volume       = {50},
  year         = {2022},
}

@article{12148,
  abstract     = {We prove a general local law for Wigner matrices that optimally handles observables of arbitrary rank and thus unifies the well-known averaged and isotropic local laws. As an application, we prove a central limit theorem in quantum unique ergodicity (QUE): that is, we show that the quadratic forms of a general deterministic matrix A on the bulk eigenvectors of a Wigner matrix have approximately Gaussian fluctuation. For the bulk spectrum, we thus generalise our previous result [17] as valid for test matrices A of large rank as well as the result of Benigni and Lopatto [7] as valid for specific small-rank observables.},
  author       = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
  issn         = {2050-5094},
  journal      = {Forum of Mathematics, Sigma},
  keywords     = {Computational Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics, Statistics and Probability, Algebra and Number Theory, Theoretical Computer Science, Analysis},
  publisher    = {Cambridge University Press},
  title        = {{Rank-uniform local law for Wigner matrices}},
  doi          = {10.1017/fms.2022.86},
  volume       = {10},
  year         = {2022},
}

@article{12179,
  abstract     = {We derive an accurate lower tail estimate on the lowest singular value σ1(X−z) of a real Gaussian (Ginibre) random matrix X shifted by a complex parameter z. Such shift effectively changes the upper tail behavior of the condition number κ(X−z) from the slower (κ(X−z)≥t)≲1/t decay typical for real Ginibre matrices to the faster 1/t2 decay seen for complex Ginibre matrices as long as z is away from the real axis. This sharpens and resolves a recent conjecture in [J. Banks et al., https://arxiv.org/abs/2005.08930, 2020] on the regularizing effect of the real Ginibre ensemble with a genuinely complex shift. As a consequence we obtain an improved upper bound on the eigenvalue condition numbers (known also as the eigenvector overlaps) for real Ginibre matrices. The main technical tool is a rigorous supersymmetric analysis from our earlier work [Probab. Math. Phys., 1 (2020), pp. 101--146].},
  author       = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
  issn         = {1095-7162},
  journal      = {SIAM Journal on Matrix Analysis and Applications},
  keywords     = {Analysis},
  number       = {3},
  pages        = {1469--1487},
  publisher    = {Society for Industrial and Applied Mathematics},
  title        = {{On the condition number of the shifted real Ginibre ensemble}},
  doi          = {10.1137/21m1424408},
  volume       = {43},
  year         = {2022},
}

@article{12232,
  abstract     = {We derive a precise asymptotic formula for the density of the small singular values of the real Ginibre matrix ensemble shifted by a complex parameter z as the dimension tends to infinity. For z away from the real axis the formula coincides with that for the complex Ginibre ensemble we derived earlier in Cipolloni et al. (Prob Math Phys 1:101–146, 2020). On the level of the one-point function of the low lying singular values we thus confirm the transition from real to complex Ginibre ensembles as the shift parameter z becomes genuinely complex; the analogous phenomenon has been well known for eigenvalues. We use the superbosonization formula (Littelmann et al. in Comm Math Phys 283:343–395, 2008) in a regime where the main contribution comes from a three dimensional saddle manifold.},
  author       = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
  issn         = {1424-0661},
  journal      = {Annales Henri Poincaré},
  keywords     = {Mathematical Physics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics},
  number       = {11},
  pages        = {3981--4002},
  publisher    = {Springer Nature},
  title        = {{Density of small singular values of the shifted real Ginibre ensemble}},
  doi          = {10.1007/s00023-022-01188-8},
  volume       = {23},
  year         = {2022},
}

@article{12243,
  abstract     = {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)]. },
  author       = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J and Xu, Yuanyuan},
  issn         = {1089-7658},
  journal      = {Journal of Mathematical Physics},
  keywords     = {Mathematical Physics, Statistical and Nonlinear Physics},
  number       = {10},
  publisher    = {AIP Publishing},
  title        = {{Directional extremal statistics for Ginibre eigenvalues}},
  doi          = {10.1063/5.0104290},
  volume       = {63},
  year         = {2022},
}

@article{12290,
  abstract     = {We prove local laws, i.e. optimal concentration estimates for arbitrary products of resolvents of a Wigner random matrix with deterministic matrices in between. We find that the size of such products heavily depends on whether some of the deterministic matrices are traceless. Our estimates correctly account for this dependence and they hold optimally down to the smallest possible spectral scale.},
  author       = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
  issn         = {1083-6489},
  journal      = {Electronic Journal of Probability},
  keywords     = {Statistics, Probability and Uncertainty, Statistics and Probability},
  pages        = {1--38},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{Optimal multi-resolvent local laws for Wigner matrices}},
  doi          = {10.1214/22-ejp838},
  volume       = {27},
  year         = {2022},
}

@article{8601,
  abstract     = {We consider large non-Hermitian real or complex random matrices X with independent, identically distributed centred entries. We prove that their local eigenvalue statistics near the spectral edge, the unit circle, coincide with those of the Ginibre ensemble, i.e. when the matrix elements of X are Gaussian. This result is the non-Hermitian counterpart of the universality of the Tracy–Widom distribution at the spectral edges of the Wigner ensemble.},
  author       = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
  issn         = {14322064},
  journal      = {Probability Theory and Related Fields},
  publisher    = {Springer Nature},
  title        = {{Edge universality for non-Hermitian random matrices}},
  doi          = {10.1007/s00440-020-01003-7},
  year         = {2021},
}

@article{9412,
  abstract     = {We extend our recent result [22] on the central limit theorem for the linear eigenvalue statistics of non-Hermitian matrices X with independent, identically distributed complex entries to the real symmetry class. We find that the expectation and variance substantially differ from their complex counterparts, reflecting (i) the special spectral symmetry of real matrices onto the real axis; and (ii) the fact that real i.i.d. matrices have many real eigenvalues. Our result generalizes the previously known special cases where either the test function is analytic [49] or the first four moments of the matrix elements match the real Gaussian [59, 44]. The key element of the proof is the analysis of several weakly dependent Dyson Brownian motions (DBMs). The conceptual novelty of the real case compared with [22] is that the correlation structure of the stochastic differentials in each individual DBM is non-trivial, potentially even jeopardising its well-posedness.},
  author       = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
  issn         = {10836489},
  journal      = {Electronic Journal of Probability},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{Fluctuation around the circular law for random matrices with real entries}},
  doi          = {10.1214/21-EJP591},
  volume       = {26},
  year         = {2021},
}

@article{10221,
  abstract     = {We prove that any deterministic matrix is approximately the identity in the eigenbasis of a large random Wigner matrix with very high probability and with an optimal error inversely proportional to the square root of the dimension. Our theorem thus rigorously verifies the Eigenstate Thermalisation Hypothesis by Deutsch (Phys Rev A 43:2046–2049, 1991) for the simplest chaotic quantum system, the Wigner ensemble. In mathematical terms, we prove the strong form of Quantum Unique Ergodicity (QUE) with an optimal convergence rate for all eigenvectors simultaneously, generalizing previous probabilistic QUE results in Bourgade and Yau (Commun Math Phys 350:231–278, 2017) and Bourgade et al. (Commun Pure Appl Math 73:1526–1596, 2020).},
  author       = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
  issn         = {1432-0916},
  journal      = {Communications in Mathematical Physics},
  number       = {2},
  pages        = {1005–1048},
  publisher    = {Springer Nature},
  title        = {{Eigenstate thermalization hypothesis for Wigner matrices}},
  doi          = {10.1007/s00220-021-04239-z},
  volume       = {388},
  year         = {2021},
}

@article{6184,
  abstract     = {We prove edge universality for a general class of correlated real symmetric or complex Hermitian Wigner matrices with arbitrary expectation. Our theorem also applies to internal edges of the self-consistent density of states. In particular, we establish a strong form of band rigidity which excludes mismatches between location and label of eigenvalues close to internal edges in these general models.},
  author       = {Alt, Johannes and Erdös, László and Krüger, Torben H and Schröder, Dominik J},
  issn         = {0091-1798},
  journal      = {Annals of Probability},
  number       = {2},
  pages        = {963--1001},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{Correlated random matrices: Band rigidity and edge universality}},
  doi          = {10.1214/19-AOP1379},
  volume       = {48},
  year         = {2020},
}

@article{6185,
  abstract     = {For complex Wigner-type matrices, i.e. Hermitian random matrices with independent, not necessarily identically distributed entries above the diagonal, we show that at any cusp singularity of the limiting eigenvalue distribution the local eigenvalue statistics are universal and form a Pearcey process. Since the density of states typically exhibits only square root or cubic root cusp singularities, our work complements previous results on the bulk and edge universality and it thus completes the resolution of the Wigner–Dyson–Mehta universality conjecture for the last remaining universality type in the complex Hermitian class. Our analysis holds not only for exact cusps, but approximate cusps as well, where an extended Pearcey process emerges. As a main technical ingredient we prove an optimal local law at the cusp for both symmetry classes. This result is also the key input in the companion paper (Cipolloni et al. in Pure Appl Anal, 2018. arXiv:1811.04055) where the cusp universality for real symmetric Wigner-type matrices is proven. The novel cusp fluctuation mechanism is also essential for the recent results on the spectral radius of non-Hermitian random matrices (Alt et al. in Spectral radius of random matrices with independent entries, 2019. arXiv:1907.13631), and the non-Hermitian edge universality (Cipolloni et al. in Edge universality for non-Hermitian random matrices, 2019. arXiv:1908.00969).},
  author       = {Erdös, László and Krüger, Torben H and Schröder, Dominik J},
  issn         = {1432-0916},
  journal      = {Communications in Mathematical Physics},
  pages        = {1203--1278},
  publisher    = {Springer Nature},
  title        = {{Cusp universality for random matrices I: Local law and the complex Hermitian case}},
  doi          = {10.1007/s00220-019-03657-4},
  volume       = {378},
  year         = {2020},
}

@article{15063,
  abstract     = {We consider the least singular value of a large random matrix with real or complex i.i.d. Gaussian entries shifted by a constant z∈C. We prove an optimal lower tail estimate on this singular value in the critical regime where z is around the spectral edge, thus improving the classical bound of Sankar, Spielman and Teng (SIAM J. Matrix Anal. Appl. 28:2 (2006), 446–476) for the particular shift-perturbation in the edge regime. Lacking Brézin–Hikami formulas in the real case, we rely on the superbosonization formula (Comm. Math. Phys. 283:2 (2008), 343–395).},
  author       = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
  issn         = {2690-1005},
  journal      = {Probability and Mathematical Physics},
  keywords     = {General Medicine},
  number       = {1},
  pages        = {101--146},
  publisher    = {Mathematical Sciences Publishers},
  title        = {{Optimal lower bound on the least singular value of the shifted Ginibre ensemble}},
  doi          = {10.2140/pmp.2020.1.101},
  volume       = {1},
  year         = {2020},
}

@phdthesis{6179,
  abstract     = {In the first part of this thesis we consider large random matrices with arbitrary expectation and a general slowly decaying correlation among its entries. We prove universality of the local eigenvalue statistics and optimal local laws for the resolvent in the bulk and edge regime. The main novel tool is a systematic diagrammatic control of a multivariate cumulant expansion.
In the second part we consider Wigner-type matrices and show that at any cusp singularity of the limiting eigenvalue distribution the local eigenvalue statistics are uni- versal and form a Pearcey process. Since the density of states typically exhibits only square root or cubic root cusp singularities, our work complements previous results on the bulk and edge universality and it thus completes the resolution of the Wigner- Dyson-Mehta universality conjecture for the last remaining universality type. Our analysis holds not only for exact cusps, but approximate cusps as well, where an ex- tended Pearcey process emerges. As a main technical ingredient we prove an optimal local law at the cusp, and extend the fast relaxation to equilibrium of the Dyson Brow- nian motion to the cusp regime.
In the third and final part we explore the entrywise linear statistics of Wigner ma- trices and identify the fluctuations for a large class of test functions with little regularity. This enables us to study the rectangular Young diagram obtained from the interlacing eigenvalues of the random matrix and its minor, and we find that, despite having the same limit, the fluctuations differ from those of the algebraic Young tableaux equipped with the Plancharel measure.},
  author       = {Schröder, Dominik J},
  issn         = {2663-337X},
  pages        = {375},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{From Dyson to Pearcey: Universal statistics in random matrix theory}},
  doi          = {10.15479/AT:ISTA:th6179},
  year         = {2019},
}