@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},
publisher = {World Scientific Publishing},
title = {{Fluctuations for differences of linear eigenvalue statistics for sample covariance matrices}},
doi = {10.1142/S2010326320500069},
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 = {Project Euclid},
title = {{Local single ring theorem on optimal scale}},
doi = {10.1214/18-AOP1284},
volume = {47},
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},
title = {{Location of the spectrum of Kronecker random matrices}},
doi = {10.1214/18-AIHP894},
volume = {55},
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},
}
@unpublished{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 used in the companion paper
[arXiv:1811.04055] where the cusp universality for real symmetric Wigner-type
matrices is proven.},
author = {Erdös, László and Krüger, Torben H and Schröder, Dominik J},
booktitle = {arXiv:1809.03971},
pages = {50},
title = {{Cusp universality for random matrices I: Local law and the complex hermitian case}},
year = {2018},
}
@unpublished{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},
booktitle = {arXiv:1811.04055},
pages = {60},
title = {{Cusp universality for random matrices II: The real symmetric case}},
year = {2018},
}
@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},
journal = {Probability Theory and Related Fields},
publisher = {Springer},
title = {{Stability of the matrix Dyson equation and random matrices with correlations}},
doi = {10.1007/s00440-018-0835-z},
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{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:1804.07752},
pages = {72},
title = {{The Dyson equation with linear self-energy: Spectral bands, edges and cusps}},
year = {2018},
}
@unpublished{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},
booktitle = {arXiv:1804.07744},
pages = {26},
title = {{Correlated random matrices: Band rigidity and edge universality}},
year = {2018},
}
@article{1337,
abstract = {We consider the local eigenvalue distribution of large self-adjoint N×N random matrices H=H∗ with centered independent entries. In contrast to previous works the matrix of variances sij=\mathbbmE|hij|2 is not assumed to be stochastic. Hence the density of states is not the Wigner semicircle law. Its possible shapes are described in the companion paper (Ajanki et al. in Quadratic Vector Equations on the Complex Upper Half Plane. arXiv:1506.05095). We show that as N grows, the resolvent, G(z)=(H−z)−1, converges to a diagonal matrix, diag(m(z)), where m(z)=(m1(z),…,mN(z)) solves the vector equation −1/mi(z)=z+∑jsijmj(z) that has been analyzed in Ajanki et al. (Quadratic Vector Equations on the Complex Upper Half Plane. arXiv:1506.05095). We prove a local law down to the smallest spectral resolution scale, and bulk universality for both real symmetric and complex hermitian symmetry classes.},
author = {Ajanki, Oskari H and Erdös, László and Krüger, Torben H},
issn = {01788051},
journal = {Probability Theory and Related Fields},
number = {3-4},
pages = {667 -- 727},
publisher = {Springer},
title = {{Universality for general Wigner-type matrices}},
doi = {10.1007/s00440-016-0740-2},
volume = {169},
year = {2017},
}
@article{1010,
abstract = {We prove a local law in the bulk of the spectrum for random Gram matrices XX∗, a generalization of sample covariance matrices, where X is a large matrix with independent, centered entries with arbitrary variances. The limiting eigenvalue density that generalizes the Marchenko-Pastur law is determined by solving a system of nonlinear equations. Our entrywise and averaged local laws are on the optimal scale with the optimal error bounds. They hold both in the square case (hard edge) and in the properly rectangular case (soft edge). In the latter case we also establish a macroscopic gap away from zero in the spectrum of XX∗. },
author = {Alt, Johannes and Erdös, László and Krüger, Torben H},
issn = {10836489},
journal = {Electronic Journal of Probability},
publisher = {Institute of Mathematical Statistics},
title = {{Local law for random Gram matrices}},
doi = {10.1214/17-EJP42},
volume = {22},
year = {2017},
}
@article{1528,
abstract = {We consider N×N Hermitian random matrices H consisting of blocks of size M≥N6/7. The matrix elements are i.i.d. within the blocks, close to a Gaussian in the four moment matching sense, but their distribution varies from block to block to form a block-band structure, with an essential band width M. We show that the entries of the Green’s function G(z)=(H−z)−1 satisfy the local semicircle law with spectral parameter z=E+iη down to the real axis for any η≫N−1, using a combination of the supersymmetry method inspired by Shcherbina (J Stat Phys 155(3): 466–499, 2014) and the Green’s function comparison strategy. Previous estimates were valid only for η≫M−1. The new estimate also implies that the eigenvectors in the middle of the spectrum are fully delocalized.},
author = {Bao, Zhigang and Erdös, László},
issn = {01788051},
journal = {Probability Theory and Related Fields},
number = {3-4},
pages = {673 -- 776},
publisher = {Springer},
title = {{Delocalization for a class of random block band matrices}},
doi = {10.1007/s00440-015-0692-y},
volume = {167},
year = {2017},
}
@book{567,
abstract = {This book is a concise and self-contained introduction of recent techniques to prove local spectral universality for large random matrices. Random matrix theory is a fast expanding research area, and this book mainly focuses on the methods that the authors participated in developing over the past few years. Many other interesting topics are not included, and neither are several new developments within the framework of these methods. The authors have chosen instead to present key concepts that they believe are the core of these methods and should be relevant for future applications. They keep technicalities to a minimum to make the book accessible to graduate students. With this in mind, they include in this book the basic notions and tools for high-dimensional analysis, such as large deviation, entropy, Dirichlet form, and the logarithmic Sobolev inequality.
},
author = {Erdös, László and Yau, Horng},
isbn = {9781470436483},
pages = {226},
publisher = {American Mathematical Society},
title = {{A dynamical approach to random matrix theory}},
volume = {28},
year = {2017},
}
@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},
}
@article{1207,
abstract = {The eigenvalue distribution of the sum of two large Hermitian matrices, when one of them is conjugated by a Haar distributed unitary matrix, is asymptotically given by the free convolution of their spectral distributions. We prove that this convergence also holds locally in the bulk of the spectrum, down to the optimal scales larger than the eigenvalue spacing. The corresponding eigenvectors are fully delocalized. Similar results hold for the sum of two real symmetric matrices, when one is conjugated by Haar orthogonal matrix.},
author = {Bao, Zhigang and Erdös, László and Schnelli, Kevin},
issn = {00103616},
journal = {Communications in Mathematical Physics},
number = {3},
pages = {947 -- 990},
publisher = {Springer},
title = {{Local law of addition of random matrices on optimal scale}},
doi = {10.1007/s00220-016-2805-6},
volume = {349},
year = {2017},
}
@article{1144,
abstract = {We show that matrix elements of functions of N × N Wigner matrices fluctuate on a scale of order N−1/2 and we identify the limiting fluctuation. Our result holds for any function f of the matrix that has bounded variation thus considerably relaxing the regularity requirement imposed in [7, 11].},
author = {Erdös, László and Schröder, Dominik J},
journal = {Electronic Communications in Probability},
publisher = {Institute of Mathematical Statistics},
title = {{Fluctuations of functions of Wigner matrices}},
doi = {10.1214/16-ECP38},
volume = {21},
year = {2017},
}