@article{8373,
abstract = {It is well known that special Kubo-Ando operator means admit divergence center interpretations, moreover, they are also mean squared error estimators for certain metrics on positive definite operators. In this paper we give a divergence center interpretation for every symmetric Kubo-Ando mean. This characterization of the symmetric means naturally leads to a definition of weighted and multivariate versions of a large class of symmetric Kubo-Ando means. We study elementary properties of these weighted multivariate means, and note in particular that in the special case of the geometric mean we recover the weighted A#H-mean introduced by Kim, Lawson, and Lim.},
author = {Pitrik, József and Virosztek, Daniel},
issn = {0024-3795},
journal = {Linear Algebra and its Applications},
keywords = {Kubo-Ando mean, weighted multivariate mean, barycenter},
pages = {203--217},
publisher = {Elsevier},
title = {{A divergence center interpretation of general symmetric Kubo-Ando means, and related weighted multivariate operator means}},
doi = {10.1016/j.laa.2020.09.007},
volume = {609},
year = {2021},
}
@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{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{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{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},
}