TY - JOUR
AB - 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.
AU - Cipolloni, Giorgio
AU - Erdös, László
ID - 6488
IS - 3
JF - Random Matrices: Theory and Application
SN - 20103263
TI - Fluctuations for differences of linear eigenvalue statistics for sample covariance matrices
VL - 9
ER -
TY - JOUR
AB - We consider the free additive convolution of two probability measures μ and ν on the real line and show that μ ⊞ v is supported on a single interval if μ and ν each has single interval support. Moreover, the density of μ ⊞ ν is proven to vanish as a square root near the edges of its support if both μ and ν have power law behavior with exponents between −1 and 1 near their edges. In particular, these results show the ubiquity of the conditions in our recent work on optimal local law at the spectral edges for addition of random matrices [5].
AU - Bao, Zhigang
AU - Erdös, László
AU - Schnelli, Kevin
ID - 9104
JF - Journal d'Analyse Mathematique
SN - 00217670
TI - On the support of the free additive convolution
VL - 142
ER -
TY - CONF
AB - We study edge asymptotics of poissonized Plancherel-type measures on skew Young diagrams (integer partitions). These measures can be seen as generalizations of those studied by Baik--Deift--Johansson and Baik--Rains in resolving Ulam's problem on longest increasing subsequences of random permutations and the last passage percolation (corner growth) discrete versions thereof. Moreover they interpolate between said measures and the uniform measure on partitions. In the new KPZ-like 1/3 exponent edge scaling limit with logarithmic corrections, we find new probability distributions generalizing the classical Tracy--Widom GUE, GOE and GSE distributions from the theory of random matrices.
AU - Betea, Dan
AU - Bouttier, Jérémie
AU - Nejjar, Peter
AU - Vuletíc, Mirjana
ID - 8175
T2 - Proceedings on the 31st International Conference on Formal Power Series and Algebraic Combinatorics
TI - New edge asymptotics of skew Young diagrams via free boundaries
ER -
TY - CONF
AB - The aim of this short note is to expound one particular issue that was discussed during the talk [10] given at the symposium ”Researches on isometries as preserver problems and related topics” at Kyoto RIMS. That is, the role of Dirac masses by describing the isometry group of various metric spaces of probability measures. This article is of survey character, and it does not contain any essentially new results.From an isometric point of view, in some cases, metric spaces of measures are similar to C(K)-type function spaces. Similarity means here that their isometries are driven by some nice transformations of the underlying space. Of course, it depends on the particular choice of the metric how nice these transformations should be. Sometimes, as we will see, being a homeomorphism is enough to generate an isometry. But sometimes we need more: the transformation must preserve the underlying distance as well. Statements claiming that isometries in questions are necessarily induced by homeomorphisms are called Banach-Stone-type results, while results asserting that the underlying transformation is necessarily an isometry are termed as isometric rigidity results.As Dirac masses can be considered as building bricks of the set of all Borel measures, a natural question arises:Is it enough to understand how an isometry acts on the set of Dirac masses? Does this action extend uniquely to all measures?In what follows, we will thoroughly investigate this question.
AU - Geher, Gyorgy Pal
AU - Titkos, Tamas
AU - Virosztek, Daniel
ID - 7035
T2 - Kyoto RIMS Kôkyûroku
TI - Dirac masses and isometric rigidity
VL - 2125
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 study effects of a bounded and compactly supported perturbation on multidimensional continuum random Schrödinger operators in the region of complete localisation. Our main emphasis is on Anderson orthogonality for random Schrödinger operators. Among others, we prove that Anderson orthogonality does occur for Fermi energies in the region of complete localisation with a non-zero probability. This partially confirms recent non-rigorous findings [V. Khemani et al., Nature Phys. 11 (2015), 560–565]. The spectral shift function plays an important role in our analysis of Anderson orthogonality. We identify it with the index of the corresponding pair of spectral projections and explore the consequences thereof. All our results rely on the main technical estimate of this paper which guarantees separate exponential decay of the disorder-averaged Schatten p-norm of χa(f(H)−f(Hτ))χb in a and b. Here, Hτ is a perturbation of the random Schrödinger operator H, χa is the multiplication operator corresponding to the indicator function of a unit cube centred about a∈Rd, and f is in a suitable class of functions of bounded variation with distributional derivative supported in the region of complete localisation for H.
AU - Dietlein, Adrian M
AU - Gebert, Martin
AU - Müller, Peter
ID - 10879
IS - 3
JF - Journal of Spectral Theory
KW - Random Schrödinger operators
KW - spectral shift function
KW - Anderson orthogonality
SN - 1664-039X
TI - Perturbations of continuum random Schrödinger operators with applications to Anderson orthogonality and the spectral shift function
VL - 9
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 -
TY - JOUR
AB - 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.
AU - Alt, Johannes
AU - Erdös, László
AU - Krüger, Torben H
AU - Nemish, Yuriy
ID - 6240
IS - 2
JF - Annales de l'institut Henri Poincare
SN - 02460203
TI - Location of the spectrum of Kronecker random matrices
VL - 55
ER -
TY - JOUR
AB - 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).
AU - Bao, Zhigang
AU - Erdös, László
AU - Schnelli, Kevin
ID - 6511
IS - 3
JF - Annals of Probability
SN - 00911798
TI - Local single ring theorem on optimal scale
VL - 47
ER -
TY - JOUR
AB - 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