TY - JOUR
AB - 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.
AU - Pitrik, József
AU - Virosztek, Daniel
ID - 8373
JF - Linear Algebra and its Applications
KW - Kubo-Ando mean
KW - weighted multivariate mean
KW - barycenter
SN - 0024-3795
TI - A divergence center interpretation of general symmetric Kubo-Ando means, and related weighted multivariate operator means
VL - 609
ER -
TY - JOUR
AB - 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.
AU - Cipolloni, Giorgio
AU - Erdös, László
AU - Schröder, Dominik J
ID - 8601
JF - Probability Theory and Related Fields
SN - 01788051
TI - Edge universality for non-Hermitian random matrices
ER -
TY - JOUR
AB - We consider the totally asymmetric simple exclusion process (TASEP) with non-random initial condition having density ρ on ℤ− and λ on ℤ+, and a second class particle initially at the origin. For ρ<λ, there is a shock and the second class particle moves with speed 1−λ−ρ. For large time t, we show that the position of the second class particle fluctuates on a t1/3 scale and determine its limiting law. We also obtain the limiting distribution of the number of steps made by the second class particle until time t.
AU - Ferrari, Patrick
AU - Ghosal, Promit
AU - Nejjar, Peter
ID - 72
IS - 3
JF - Annales de l'institut Henri Poincare (B) Probability and Statistics
SN - 02460203
TI - Limit law of a second class particle in TASEP with non-random initial condition
VL - 55
ER -
TY - JOUR
AB - We compare finite rank perturbations of the following three ensembles of complex rectangular random matrices: First, a generalised Wishart ensemble with one random and two fixed correlation matrices introduced by Borodin and Péché, second, the product of two independent random matrices where one has correlated entries, and third, the case when the two random matrices become also coupled through a fixed matrix. The singular value statistics of all three ensembles is shown to be determinantal and we derive double contour integral representations for their respective kernels. Three different kernels are found in the limit of infinite matrix dimension at the origin of the spectrum. They depend on finite rank perturbations of the correlation and coupling matrices and are shown to be integrable. The first kernel (I) is found for two independent matrices from the second, and two weakly coupled matrices from the third ensemble. It generalises the Meijer G-kernel for two independent and uncorrelated matrices. The third kernel (III) is obtained for the generalised Wishart ensemble and for two strongly coupled matrices. It further generalises the perturbed Bessel kernel of Desrosiers and Forrester. Finally, kernel (II), found for the ensemble of two coupled matrices, provides an interpolation between the kernels (I) and (III), generalising previous findings of part of the authors.
AU - Akemann, Gernot
AU - Checinski, Tomasz
AU - Liu, Dangzheng
AU - Strahov, Eugene
ID - 7423
IS - 1
JF - Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
SN - 0246-0203
TI - Finite rank perturbations in products of coupled random matrices: From one correlated to two Wishart ensembles
VL - 55
ER -
TY - JOUR
AB - 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.
AU - Virosztek, Daniel
ID - 405
JF - Linear Algebra and Its Applications
TI - Jointly convex quantum Jensen divergences
VL - 576
ER -
TY - JOUR
AB - 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.
AU - Ajanki, Oskari H
AU - Erdös, László
AU - Krüger, Torben H
ID - 429
IS - 1-2
JF - Probability Theory and Related Fields
SN - 01788051
TI - Stability of the matrix Dyson equation and random matrices with correlations
VL - 173
ER -
TY - JOUR
AB - 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.
AU - Sadel, Christian
AU - Xu, Disheng
ID - 6086
IS - 4
JF - Ergodic Theory and Dynamical Systems
TI - Singular analytic linear cocycles with negative infinite Lyapunov exponents
VL - 39
ER -
TY - THES
AB - In the first part of this thesis we consider large random matrices with arbitrary expectation and a general slowly decaying correlation among its entries. We prove universality of the local eigenvalue statistics and optimal local laws for the resolvent in the bulk and edge regime. The main novel tool is a systematic diagrammatic control of a multivariate cumulant expansion.
In the second part we consider Wigner-type matrices and show that at any cusp singularity of the limiting eigenvalue distribution the local eigenvalue statistics are uni- versal 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. Our analysis holds not only for exact cusps, but approximate cusps as well, where an ex- tended Pearcey process emerges. As a main technical ingredient we prove an optimal local law at the cusp, and extend the fast relaxation to equilibrium of the Dyson Brow- nian motion to the cusp regime.
In the third and final part we explore the entrywise linear statistics of Wigner ma- trices and identify the fluctuations for a large class of test functions with little regularity. This enables us to study the rectangular Young diagram obtained from the interlacing eigenvalues of the random matrix and its minor, and we find that, despite having the same limit, the fluctuations differ from those of the algebraic Young tableaux equipped with the Plancharel measure.
AU - Schröder, Dominik J
ID - 6179
TI - From Dyson to Pearcey: Universal statistics in random matrix theory
ER -
TY - JOUR
AB - 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.
AU - Erdös, László
AU - Krüger, Torben H
AU - Schröder, Dominik J
ID - 6182
JF - Forum of Mathematics, Sigma
TI - Random matrices with slow correlation decay
VL - 7
ER -
TY - JOUR
AB - 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.
AU - Cipolloni, Giorgio
AU - Erdös, László
AU - Krüger, Torben H
AU - Schröder, Dominik J
ID - 6186
IS - 4
JF - Pure and Applied Analysis
SN - 2578-5893
TI - Cusp universality for random matrices, II: The real symmetric case
VL - 1
ER -