@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{9550,
abstract = {We prove that the energy of any eigenvector of a sum of several independent large Wigner matrices is equally distributed among these matrices with very high precision. This shows a particularly strong microcanonical form of the equipartition principle for quantum systems whose components are modelled by Wigner matrices. },
author = {Bao, Zhigang and Erdös, László and Schnelli, Kevin},
issn = {20505094},
journal = {Forum of Mathematics, Sigma},
publisher = {Cambridge University Press},
title = {{Equipartition principle for Wigner matrices}},
doi = {10.1017/fms.2021.38},
volume = {9},
year = {2021},
}
@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 = {2020},
}
@article{7512,
abstract = {We consider general self-adjoint polynomials in several independent random matrices whose entries are centered and have the same variance. We show that under certain conditions the local law holds up to the optimal scale, i.e., the eigenvalue density on scales just above the eigenvalue spacing follows the global density of states which is determined by free probability theory. We prove that these conditions hold for general homogeneous polynomials of degree two and for symmetrized products of independent matrices with i.i.d. entries, thus establishing the optimal bulk local law for these classes of ensembles. In particular, we generalize a similar result of Anderson for anticommutator. For more general polynomials our conditions are effectively checkable numerically.},
author = {Erdös, László and Krüger, Torben H and Nemish, Yuriy},
issn = {10960783},
journal = {Journal of Functional Analysis},
number = {12},
publisher = {Elsevier},
title = {{Local laws for polynomials of Wigner matrices}},
doi = {10.1016/j.jfa.2020.108507},
volume = {278},
year = {2020},
}
@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},
journal = {Annals of Probability},
number = {2},
pages = {963--1001},
publisher = {Project Euclid},
title = {{Correlated random matrices: Band rigidity and edge universality}},
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{6488,
abstract = {We prove a central limit theorem for the difference of linear eigenvalue statistics of a sample covariance matrix W˜ and its minor W. We find that the fluctuation of this difference is much smaller than those of the individual linear statistics, as a consequence of the strong correlation between the eigenvalues of W˜ and W. Our result identifies the fluctuation of the spatial derivative of the approximate Gaussian field in the recent paper by Dumitru and Paquette. Unlike in a similar result for Wigner matrices, for sample covariance matrices, the fluctuation may entirely vanish.},
author = {Cipolloni, Giorgio and Erdös, László},
issn = {20103271},
journal = {Random Matrices: Theory and Application},
number = {3},
publisher = {World Scientific Publishing},
title = {{Fluctuations for differences of linear eigenvalue statistics for sample covariance matrices}},
doi = {10.1142/S2010326320500069},
volume = {9},
year = {2020},
}
@article{9104,
abstract = {We consider the free additive convolution of two probability measures μ and ν on the real line and show that μ ⊞ v is supported on a single interval if μ and ν each has single interval support. Moreover, the density of μ ⊞ ν is proven to vanish as a square root near the edges of its support if both μ and ν have power law behavior with exponents between −1 and 1 near their edges. In particular, these results show the ubiquity of the conditions in our recent work on optimal local law at the spectral edges for addition of random matrices [5].},
author = {Bao, Zhigang and Erdös, László and Schnelli, Kevin},
issn = {15658538},
journal = {Journal d'Analyse Mathematique},
pages = {323--348},
publisher = {Springer Nature},
title = {{On the support of the free additive convolution}},
doi = {10.1007/s11854-020-0135-2},
volume = {142},
year = {2020},
}
@article{6182,
abstract = {We consider large random matrices with a general slowly decaying correlation among its entries. We prove universality of the local eigenvalue statistics and optimal local laws for the resolvent away from the spectral edges, generalizing the recent result of Ajanki et al. [‘Stability of the matrix Dyson equation and random matrices with correlations’, Probab. Theory Related Fields 173(1–2) (2019), 293–373] to allow slow correlation decay and arbitrary expectation. The main novel tool is
a systematic diagrammatic control of a multivariate cumulant expansion.},
author = {Erdös, László and Krüger, Torben H and Schröder, Dominik J},
issn = {20505094},
journal = {Forum of Mathematics, Sigma},
publisher = {Cambridge University Press},
title = {{Random matrices with slow correlation decay}},
doi = {10.1017/fms.2019.2},
volume = {7},
year = {2019},
}
@article{6186,
abstract = {We prove that the local eigenvalue statistics of real symmetric Wigner-type
matrices near the cusp points of the eigenvalue density are universal. Together
with the companion paper [arXiv:1809.03971], which proves the same result for
the complex Hermitian symmetry class, this completes the last remaining case of
the Wigner-Dyson-Mehta universality conjecture after bulk and edge
universalities have been established in the last years. We extend the recent
Dyson Brownian motion analysis at the edge [arXiv:1712.03881] to the cusp
regime using the optimal local law from [arXiv:1809.03971] and the accurate
local shape analysis of the density from [arXiv:1506.05095, arXiv:1804.07752].
We also present a PDE-based method to improve the estimate on eigenvalue
rigidity via the maximum principle of the heat flow related to the Dyson
Brownian motion.},
author = {Cipolloni, Giorgio and Erdös, László and Krüger, Torben H and Schröder, Dominik J},
issn = {2578-5885},
journal = {Pure and Applied Analysis },
number = {4},
pages = {615–707},
publisher = {MSP},
title = {{Cusp universality for random matrices, II: The real symmetric case}},
doi = {10.2140/paa.2019.1.615},
volume = {1},
year = {2019},
}
@article{6240,
abstract = {For a general class of large non-Hermitian random block matrices X we prove that there are no eigenvalues away from a deterministic set with very high probability. This set is obtained from the Dyson equation of the Hermitization of X as the self-consistent approximation of the pseudospectrum. We demonstrate that the analysis of the matrix Dyson equation from (Probab. Theory Related Fields (2018)) offers a unified treatment of many structured matrix ensembles.},
author = {Alt, Johannes and Erdös, László and Krüger, Torben H and Nemish, Yuriy},
issn = {02460203},
journal = {Annales de l'institut Henri Poincare},
number = {2},
pages = {661--696},
publisher = {Institut Henri Poincaré},
title = {{Location of the spectrum of Kronecker random matrices}},
doi = {10.1214/18-AIHP894},
volume = {55},
year = {2019},
}
@article{6511,
abstract = {Let U and V be two independent N by N random matrices that are distributed according to Haar measure on U(N). Let Σ be a nonnegative deterministic N by N matrix. The single ring theorem [Ann. of Math. (2) 174 (2011) 1189–1217] asserts that the empirical eigenvalue distribution of the matrix X:=UΣV∗ converges weakly, in the limit of large N, to a deterministic measure which is supported on a single ring centered at the origin in ℂ. Within the bulk regime, that is, in the interior of the single ring, we establish the convergence of the empirical eigenvalue distribution on the optimal local scale of order N−1/2+ε and establish the optimal convergence rate. The same results hold true when U and V are Haar distributed on O(N).},
author = {Bao, Zhigang and Erdös, László and Schnelli, Kevin},
issn = {00911798},
journal = {Annals of Probability},
number = {3},
pages = {1270--1334},
publisher = {Institute of Mathematical Statistics},
title = {{Local single ring theorem on optimal scale}},
doi = {10.1214/18-AOP1284},
volume = {47},
year = {2019},
}
@article{429,
abstract = {We consider real symmetric or complex hermitian random matrices with correlated entries. We prove local laws for the resolvent and universality of the local eigenvalue statistics in the bulk of the spectrum. The correlations have fast decay but are otherwise of general form. The key novelty is the detailed stability analysis of the corresponding matrix valued Dyson equation whose solution is the deterministic limit of the resolvent.},
author = {Ajanki, Oskari H and Erdös, László and Krüger, Torben H},
issn = {14322064},
journal = {Probability Theory and Related Fields},
number = {1-2},
pages = {293–373},
publisher = {Springer},
title = {{Stability of the matrix Dyson equation and random matrices with correlations}},
doi = {10.1007/s00440-018-0835-z},
volume = {173},
year = {2019},
}
@article{181,
abstract = {We consider large random matrices X with centered, independent entries but possibly di erent variances. We compute the normalized trace of f(X)g(X∗) for f, g functions analytic on the spectrum of X. We use these results to compute the long time asymptotics for systems of coupled di erential equations with random coe cients. We show that when the coupling is critical, the norm squared of the solution decays like t−1/2.},
author = {Erdös, László and Krüger, Torben H and Renfrew, David T},
journal = {SIAM Journal on Mathematical Analysis},
number = {3},
pages = {3271 -- 3290},
publisher = {Society for Industrial and Applied Mathematics },
title = {{Power law decay for systems of randomly coupled differential equations}},
doi = {10.1137/17M1143125},
volume = {50},
year = {2018},
}
@article{566,
abstract = {We consider large random matrices X with centered, independent entries which have comparable but not necessarily identical variances. Girko's circular law asserts that the spectrum is supported in a disk and in case of identical variances, the limiting density is uniform. In this special case, the local circular law by Bourgade et. al. [11,12] shows that the empirical density converges even locally on scales slightly above the typical eigenvalue spacing. In the general case, the limiting density is typically inhomogeneous and it is obtained via solving a system of deterministic equations. Our main result is the local inhomogeneous circular law in the bulk spectrum on the optimal scale for a general variance profile of the entries of X.
},
author = {Alt, Johannes and Erdös, László and Krüger, Torben H},
journal = {Annals Applied Probability },
number = {1},
pages = {148--203},
publisher = {Institute of Mathematical Statistics},
title = {{Local inhomogeneous circular law}},
doi = {10.1214/17-AAP1302},
volume = {28},
year = {2018},
}
@article{5971,
abstract = {We consider a Wigner-type ensemble, i.e. large hermitian N×N random matrices H=H∗ with centered independent entries and with a general matrix of variances Sxy=𝔼∣∣Hxy∣∣2. The norm of H is asymptotically given by the maximum of the support of the self-consistent density of states. We establish a bound on this maximum in terms of norms of powers of S that substantially improves the earlier bound 2∥S∥1/2∞ given in [O. Ajanki, L. Erdős and T. Krüger, Universality for general Wigner-type matrices, Prob. Theor. Rel. Fields169 (2017) 667–727]. The key element of the proof is an effective Markov chain approximation for the contributions of the weighted Dyck paths appearing in the iterative solution of the corresponding Dyson equation.},
author = {Erdös, László and Mühlbacher, Peter},
issn = {2010-3263},
journal = {Random matrices: Theory and applications},
publisher = {World Scientific Publishing},
title = {{Bounds on the norm of Wigner-type random matrices}},
doi = {10.1142/s2010326319500096},
year = {2018},
}
@unpublished{6183,
abstract = {We study the unique solution $m$ of the Dyson equation \[ -m(z)^{-1} = z - a
+ S[m(z)] \] on a von Neumann algebra $\mathcal{A}$ with the constraint
$\mathrm{Im}\,m\geq 0$. Here, $z$ lies in the complex upper half-plane, $a$ is
a self-adjoint element of $\mathcal{A}$ and $S$ is a positivity-preserving
linear operator on $\mathcal{A}$. We show that $m$ is the Stieltjes transform
of a compactly supported $\mathcal{A}$-valued measure on $\mathbb{R}$. Under
suitable assumptions, we establish that this measure has a uniformly
$1/3$-H\"{o}lder continuous density with respect to the Lebesgue measure, which
is supported on finitely many intervals, called bands. In fact, the density is
analytic inside the bands with a square-root growth at the edges and internal
cubic root cusps whenever the gap between two bands vanishes. The shape of
these singularities is universal and no other singularity may occur. We give a
precise asymptotic description of $m$ near the singular points. These
asymptotics generalize the analysis at the regular edges given in the companion
paper on the Tracy-Widom universality for the edge eigenvalue statistics for
correlated random matrices [arXiv:1804.07744] and they play a key role in the
proof of the Pearcey universality at the cusp for Wigner-type matrices
[arXiv:1809.03971,arXiv:1811.04055]. We also extend the finite dimensional band
mass formula from [arXiv:1804.07744] to the von Neumann algebra setting by
showing that the spectral mass of the bands is topologically rigid under
deformations and we conclude that these masses are quantized in some important
cases.},
author = {Alt, Johannes and Erdös, László and Krüger, Torben H},
booktitle = {arXiv},
title = {{The Dyson equation with linear self-energy: Spectral bands, edges and cusps}},
year = {2018},
}
@article{1012,
abstract = {We prove a new central limit theorem (CLT) for the difference of linear eigenvalue statistics of a Wigner random matrix H and its minor H and find that the fluctuation is much smaller than the fluctuations of the individual linear statistics, as a consequence of the strong correlation between the eigenvalues of H and H. In particular, our theorem identifies the fluctuation of Kerov's rectangular Young diagrams, defined by the interlacing eigenvalues ofH and H, around their asymptotic shape, the Vershik'Kerov'Logan'Shepp curve. Young diagrams equipped with the Plancherel measure follow the same limiting shape. For this, algebraically motivated, ensemble a CLT has been obtained in Ivanov and Olshanski [20] which is structurally similar to our result but the variance is different, indicating that the analogy between the two models has its limitations. Moreover, our theorem shows that Borodin's result [7] on the convergence of the spectral distribution of Wigner matrices to a Gaussian free field also holds in derivative sense.},
author = {Erdös, László and Schröder, Dominik J},
issn = {10737928},
journal = {International Mathematics Research Notices},
number = {10},
pages = {3255--3298},
publisher = {Oxford University Press},
title = {{Fluctuations of rectangular young diagrams of interlacing wigner eigenvalues}},
doi = {10.1093/imrn/rnw330},
volume = {2018},
year = {2018},
}
@article{721,
abstract = {Let S be a positivity-preserving symmetric linear operator acting on bounded functions. The nonlinear equation -1/m=z+Sm with a parameter z in the complex upper half-plane ℍ has a unique solution m with values in ℍ. We show that the z-dependence of this solution can be represented as the Stieltjes transforms of a family of probability measures v on ℝ. Under suitable conditions on S, we show that v has a real analytic density apart from finitely many algebraic singularities of degree at most 3. Our motivation comes from large random matrices. The solution m determines the density of eigenvalues of two prominent matrix ensembles: (i) matrices with centered independent entries whose variances are given by S and (ii) matrices with correlated entries with a translation-invariant correlation structure. Our analysis shows that the limiting eigenvalue density has only square root singularities or cubic root cusps; no other singularities occur.},
author = {Ajanki, Oskari H and Krüger, Torben H and Erdös, László},
issn = {00103640},
journal = {Communications on Pure and Applied Mathematics},
number = {9},
pages = {1672 -- 1705},
publisher = {Wiley-Blackwell},
title = {{Singularities of solutions to quadratic vector equations on the complex upper half plane}},
doi = {10.1002/cpa.21639},
volume = {70},
year = {2017},
}
@article{733,
abstract = {Let A and B be two N by N deterministic Hermitian matrices and let U be an N by N Haar distributed unitary matrix. It is well known that the spectral distribution of the sum H = A + UBU∗ converges weakly to the free additive convolution of the spectral distributions of A and B, as N tends to infinity. We establish the optimal convergence rate in the bulk of the spectrum.},
author = {Bao, Zhigang and Erdös, László and Schnelli, Kevin},
journal = {Advances in Mathematics},
pages = {251 -- 291},
publisher = {Academic Press},
title = {{Convergence rate for spectral distribution of addition of random matrices}},
doi = {10.1016/j.aim.2017.08.028},
volume = {319},
year = {2017},
}