@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 = {50},
publisher = {Springer Nature},
title = {{Cusp universality for random matrices I: Local law and the complex hermitian case}},
doi = {10.1007/s00220-019-03657-4},
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},
publisher = {Elsevier},
title = {{Local laws for polynomials of Wigner matrices}},
doi = {10.1016/j.jfa.2020.108507},
year = {2020},
}
@article{7618,
abstract = {This short note aims to study quantum Hellinger distances investigated recently by Bhatia et al. (Lett Math Phys 109:1777–1804, 2019) with a particular emphasis on barycenters. We introduce the family of generalized quantum Hellinger divergences that are of the form ϕ(A,B)=Tr((1−c)A+cB−AσB), where σ is an arbitrary Kubo–Ando mean, and c∈(0,1) is the weight of σ. We note that these divergences belong to the family of maximal quantum f-divergences, and hence are jointly convex, and satisfy the data processing inequality. We derive a characterization of the barycenter of finitely many positive definite operators for these generalized quantum Hellinger divergences. We note that the characterization of the barycenter as the weighted multivariate 1/2-power mean, that was claimed in Bhatia et al. (2019), is true in the case of commuting operators, but it is not correct in the general case. },
author = {Pitrik, Jozsef and Virosztek, Daniel},
issn = {1573-0530},
journal = {Letters in Mathematical Physics},
number = {8},
pages = {2039--2052},
publisher = {Springer Nature},
title = {{Quantum Hellinger distances revisited}},
doi = {10.1007/s11005-020-01282-0},
volume = {110},
year = {2020},
}
@article{7389,
abstract = {Recently Kloeckner described the structure of the isometry group of the quadratic Wasserstein space W_2(R^n). It turned out that the case of the real line is exceptional in the sense that there exists an exotic isometry flow. Following this line of investigation, we compute Isom(W_p(R)), the isometry group of the Wasserstein space
W_p(R) for all p \in [1,\infty) \setminus {2}. We show that W_2(R) is also exceptional regarding the
parameter p: W_p(R) is isometrically rigid if and only if p is not equal to 2. Regarding the underlying
space, we prove that the exceptionality of p = 2 disappears if we replace R by the compact
interval [0,1]. Surprisingly, in that case, W_p([0,1]) is isometrically rigid if and only if
p is not equal to 1. Moreover, W_1([0,1]) admits isometries that split mass, and Isom(W_1([0,1]))
cannot be embedded into Isom(W_1(R)).},
author = {Geher, Gyorgy Pal and Titkos, Tamas and Virosztek, Daniel},
issn = {10886850},
journal = {Transactions of the American Mathematical Society},
keywords = {Wasserstein space, isometric embeddings, isometric rigidity, exotic isometry flow},
number = {8},
pages = {5855--5883},
publisher = {American Mathematical Society},
title = {{Isometric study of Wasserstein spaces - the real line}},
doi = {10.1090/tran/8113},
volume = {373},
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{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{6086,
abstract = {We show that linear analytic cocycles where all Lyapunov exponents are negative infinite are nilpotent. For such one-frequency cocycles we show that they can be analytically conjugated to an upper triangular cocycle or a Jordan normal form. As a consequence, an arbitrarily small analytic perturbation leads to distinct Lyapunov exponents. Moreover, in the one-frequency case where the th Lyapunov exponent is finite and the st negative infinite, we obtain a simple criterion for domination in which case there is a splitting into a nilpotent part and an invertible part.},
author = {Sadel, Christian and Xu, Disheng},
journal = {Ergodic Theory and Dynamical Systems},
number = {4},
pages = {1082--1098},
publisher = {Cambridge University Press},
title = {{Singular analytic linear cocycles with negative infinite Lyapunov exponents}},
doi = {10.1017/etds.2017.52},
volume = {39},
year = {2019},
}
@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{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 = {Institute of Mathematical Statistics},
title = {{Local single ring theorem on optimal scale}},
doi = {10.1214/18-AOP1284},
volume = {47},
year = {2019},
}
@article{405,
abstract = {We investigate the quantum Jensen divergences from the viewpoint of joint convexity. It turns out that the set of the functions which generate jointly convex quantum Jensen divergences on positive matrices coincides with the Matrix Entropy Class which has been introduced by Chen and Tropp quite recently.},
author = {Virosztek, Daniel},
journal = {Linear Algebra and Its Applications},
pages = {67--78},
publisher = {Elsevier},
title = {{Jointly convex quantum Jensen divergences}},
doi = {10.1016/j.laa.2018.03.002},
volume = {576},
year = {2019},
}
@article{6843,
abstract = {The aim of this short paper is to offer a complete characterization of all (not necessarily surjective) isometric embeddings of the Wasserstein space Wp(X), where S is a countable discrete metric space and 0