@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},
}
@phdthesis{9022,
abstract = {In the first part of the thesis we consider Hermitian random matrices. Firstly, we consider sample covariance matrices XX∗ with X having independent identically distributed (i.i.d.) centred entries. We prove a Central Limit Theorem for differences of linear statistics of XX∗ and its minor after removing the first column of X. Secondly, we consider Wigner-type matrices and prove that the eigenvalue statistics near cusp singularities of the limiting density of states are universal and that they form a Pearcey process. Since the limiting eigenvalue distribution admits only square root (edge) and cubic root (cusp) singularities, this concludes the third and last remaining case of the Wigner-Dyson-Mehta universality conjecture. The main technical ingredients are an optimal local law at the cusp, and the proof of the fast relaxation to equilibrium of the Dyson Brownian motion in the cusp regime.
In the second part we consider non-Hermitian matrices X with centred i.i.d. entries. We normalise the entries of X to have variance N −1. It is well known that the empirical eigenvalue density converges to the uniform distribution on the unit disk (circular law). In the first project, we prove universality of the local eigenvalue statistics close to the edge of the spectrum. This is the non-Hermitian analogue of the TracyWidom universality at the Hermitian edge. Technically we analyse the evolution of the spectral distribution of X along the Ornstein-Uhlenbeck flow for very long time
(up to t = +∞). In the second project, we consider linear statistics of eigenvalues for macroscopic test functions f in the Sobolev space H2+ϵ and prove their convergence to the projection of the Gaussian Free Field on the unit disk. We prove this result for non-Hermitian matrices with real or complex entries. The main technical ingredients are: (i) local law for products of two resolvents at different spectral parameters, (ii) analysis of correlated Dyson Brownian motions.
In the third and final part we discuss the mathematically rigorous application of supersymmetric techniques (SUSY ) to give a lower tail estimate of the lowest singular value of X − z, with z ∈ C. More precisely, we use superbosonisation formula to give an integral representation of the resolvent of (X − z)(X − z)∗ which reduces to two and three contour integrals in the complex and real case, respectively. The rigorous analysis of these integrals is quite challenging since simple saddle point analysis cannot be applied (the main contribution comes from a non-trivial manifold). Our result
improves classical smoothing inequalities in the regime |z| ≈ 1; this result is essential to prove edge universality for i.i.d. non-Hermitian matrices.},
author = {Cipolloni, Giorgio},
issn = {2663-337X},
pages = {380},
publisher = {IST Austria},
title = {{Fluctuations in the spectrum of random matrices}},
doi = {10.15479/AT:ISTA:9022},
year = {2021},
}
@article{9036,
abstract = {In this short note, we prove that the square root of the quantum Jensen-Shannon divergence is a true metric on the cone of positive matrices, and hence in particular on the quantum state space.},
author = {Virosztek, Daniel},
issn = {0001-8708},
journal = {Advances in Mathematics},
keywords = {General Mathematics},
number = {3},
publisher = {Elsevier},
title = {{The metric property of the quantum Jensen-Shannon divergence}},
doi = {10.1016/j.aim.2021.107595},
volume = {380},
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{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{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{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{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},
}