@inproceedings{7035, abstract = {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.}, author = {Geher, Gyorgy Pal and Titkos, Tamas and Virosztek, Daniel}, booktitle = {Kyoto RIMS Kôkyûroku}, location = {Kyoto, Japan}, pages = {34--41}, publisher = {Research Institute for Mathematical Sciences, Kyoto University}, title = {{Dirac masses and isometric rigidity}}, volume = {2125}, year = {2019}, } @article{72, abstract = {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.}, author = {Ferrari, Patrick and Ghosal, Promit and Nejjar, Peter}, issn = {0246-0203}, journal = {Annales de l'institut Henri Poincare (B) Probability and Statistics}, number = {3}, pages = {1203--1225}, publisher = {Institute of Mathematical Statistics}, title = {{Limit law of a second class particle in TASEP with non-random initial condition}}, doi = {10.1214/18-AIHP916}, volume = {55}, year = {2019}, } @article{7423, abstract = {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.}, author = {Akemann, Gernot and Checinski, Tomasz and Liu, Dangzheng and Strahov, Eugene}, issn = {0246-0203}, journal = {Annales de l'Institut Henri Poincaré, Probabilités et Statistiques}, number = {1}, pages = {441--479}, publisher = {Institute of Mathematical Statistics}, title = {{Finite rank perturbations in products of coupled random matrices: From one correlated to two Wishart ensembles}}, doi = {10.1214/18-aihp888}, volume = {55}, 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}, } @phdthesis{6179, abstract = {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.}, author = {Schröder, Dominik J}, issn = {2663-337X}, pages = {375}, publisher = {Institute of Science and Technology Austria}, title = {{From Dyson to Pearcey: Universal statistics in random matrix theory}}, doi = {10.15479/AT:ISTA:th6179}, 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 = {0246-0203}, 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{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{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{284, abstract = {Borel probability measures living on metric spaces are fundamental mathematical objects. There are several meaningful distance functions that make the collection of the probability measures living on a certain space a metric space. We are interested in the description of the structure of the isometries of such metric spaces. We overview some of the recent results of the topic and we also provide some new ones concerning the Wasserstein distance. More specifically, we consider the space of all Borel probability measures on the unit sphere of a Euclidean space endowed with the Wasserstein metric W_p for arbitrary p >= 1, and we show that the action of a Wasserstein isometry on the set of Dirac measures is induced by an isometry of the underlying unit sphere.}, author = {Virosztek, Daniel}, issn = {2064-8316}, journal = {Acta Scientiarum Mathematicarum}, number = {1-2}, pages = {65 -- 80}, publisher = {Springer Nature}, title = {{Maps on probability measures preserving certain distances - a survey and some new results}}, doi = {10.14232/actasm-018-753-y}, volume = {84}, year = {2018}, } @article{181, abstract = {We consider large random matrices X with centered, independent entries but possibly di erent variances. We compute the normalized trace of f(X)g(X∗) for f, g functions analytic on the spectrum of X. We use these results to compute the long time asymptotics for systems of coupled di erential equations with random coe cients. We show that when the coupling is critical, the norm squared of the solution decays like t−1/2.}, author = {Erdös, László and Krüger, Torben H and Renfrew, David T}, journal = {SIAM Journal on Mathematical Analysis}, number = {3}, pages = {3271 -- 3290}, publisher = {Society for Industrial and Applied Mathematics }, title = {{Power law decay for systems of randomly coupled differential equations}}, doi = {10.1137/17M1143125}, volume = {50}, year = {2018}, } @article{690, abstract = {We consider spectral properties and the edge universality of sparse random matrices, the class of random matrices that includes the adjacency matrices of the Erdős–Rényi graph model G(N, p). We prove a local law for the eigenvalue density up to the spectral edges. Under a suitable condition on the sparsity, we also prove that the rescaled extremal eigenvalues exhibit GOE Tracy–Widom fluctuations if a deterministic shift of the spectral edge due to the sparsity is included. For the adjacency matrix of the Erdős–Rényi graph this establishes the Tracy–Widom fluctuations of the second largest eigenvalue when p is much larger than N−2/3 with a deterministic shift of order (Np)−1.}, author = {Lee, Jii and Schnelli, Kevin}, journal = {Probability Theory and Related Fields}, number = {1-2}, publisher = {Springer}, title = {{Local law and Tracy–Widom limit for sparse random matrices}}, doi = {10.1007/s00440-017-0787-8}, volume = {171}, year = {2018}, } @article{70, abstract = {We consider the totally asymmetric simple exclusion process in a critical scaling parametrized by a≥0, which creates a shock in the particle density of order aT−1/3, T the observation time. When starting from step initial data, we provide bounds on the limiting law which in particular imply that in the double limit lima→∞limT→∞ one recovers the product limit law and the degeneration of the correlation length observed at shocks of order 1. This result is shown to apply to a general last-passage percolation model. We also obtain bounds on the two-point functions of several airy processes.}, author = {Nejjar, Peter}, issn = {1980-0436}, journal = {Latin American Journal of Probability and Mathematical Statistics}, number = {2}, pages = {1311--1334}, publisher = {Instituto Nacional de Matematica Pura e Aplicada}, title = {{Transition to shocks in TASEP and decoupling of last passage times}}, doi = {10.30757/ALEA.v15-49}, volume = {15}, year = {2018}, } @article{556, abstract = {We investigate the free boundary Schur process, a variant of the Schur process introduced by Okounkov and Reshetikhin, where we allow the first and the last partitions to be arbitrary (instead of empty in the original setting). The pfaffian Schur process, previously studied by several authors, is recovered when just one of the boundary partitions is left free. We compute the correlation functions of the process in all generality via the free fermion formalism, which we extend with the thorough treatment of “free boundary states.” For the case of one free boundary, our approach yields a new proof that the process is pfaffian. For the case of two free boundaries, we find that the process is not pfaffian, but a closely related process is. We also study three different applications of the Schur process with one free boundary: fluctuations of symmetrized last passage percolation models, limit shapes and processes for symmetric plane partitions and for plane overpartitions.}, author = {Betea, Dan and Bouttier, Jeremie and Nejjar, Peter and Vuletic, Mirjana}, issn = {1424-0637}, journal = {Annales Henri Poincare}, number = {12}, pages = {3663--3742}, publisher = {Springer Nature}, title = {{The free boundary Schur process and applications I}}, doi = {10.1007/s00023-018-0723-1}, volume = {19}, year = {2018}, } @article{566, abstract = {We consider large random matrices X with centered, independent entries which have comparable but not necessarily identical variances. Girko's circular law asserts that the spectrum is supported in a disk and in case of identical variances, the limiting density is uniform. In this special case, the local circular law by Bourgade et. al. [11,12] shows that the empirical density converges even locally on scales slightly above the typical eigenvalue spacing. In the general case, the limiting density is typically inhomogeneous and it is obtained via solving a system of deterministic equations. Our main result is the local inhomogeneous circular law in the bulk spectrum on the optimal scale for a general variance profile of the entries of X. }, author = {Alt, Johannes and Erdös, László and Krüger, Torben H}, journal = {Annals Applied Probability }, number = {1}, pages = {148--203}, publisher = {Institute of Mathematical Statistics}, title = {{Local inhomogeneous circular law}}, doi = {10.1214/17-AAP1302}, volume = {28}, year = {2018}, } @article{5971, abstract = {We consider a Wigner-type ensemble, i.e. large hermitian N×N random matrices H=H∗ with centered independent entries and with a general matrix of variances Sxy=𝔼∣∣Hxy∣∣2. The norm of H is asymptotically given by the maximum of the support of the self-consistent density of states. We establish a bound on this maximum in terms of norms of powers of S that substantially improves the earlier bound 2∥S∥1/2∞ given in [O. Ajanki, L. Erdős and T. Krüger, Universality for general Wigner-type matrices, Prob. Theor. Rel. Fields169 (2017) 667–727]. The key element of the proof is an effective Markov chain approximation for the contributions of the weighted Dyck paths appearing in the iterative solution of the corresponding Dyson equation.}, author = {Erdös, László and Mühlbacher, Peter}, issn = {2010-3271}, journal = {Random matrices: Theory and applications}, publisher = {World Scientific Publishing}, title = {{Bounds on the norm of Wigner-type random matrices}}, doi = {10.1142/s2010326319500096}, year = {2018}, } @unpublished{6183, abstract = {We study the unique solution $m$ of the Dyson equation \[ -m(z)^{-1} = z - a + S[m(z)] \] on a von Neumann algebra $\mathcal{A}$ with the constraint $\mathrm{Im}\,m\geq 0$. Here, $z$ lies in the complex upper half-plane, $a$ is a self-adjoint element of $\mathcal{A}$ and $S$ is a positivity-preserving linear operator on $\mathcal{A}$. We show that $m$ is the Stieltjes transform of a compactly supported $\mathcal{A}$-valued measure on $\mathbb{R}$. Under suitable assumptions, we establish that this measure has a uniformly $1/3$-H\"{o}lder continuous density with respect to the Lebesgue measure, which is supported on finitely many intervals, called bands. In fact, the density is analytic inside the bands with a square-root growth at the edges and internal cubic root cusps whenever the gap between two bands vanishes. The shape of these singularities is universal and no other singularity may occur. We give a precise asymptotic description of $m$ near the singular points. These asymptotics generalize the analysis at the regular edges given in the companion paper on the Tracy-Widom universality for the edge eigenvalue statistics for correlated random matrices [arXiv:1804.07744] and they play a key role in the proof of the Pearcey universality at the cusp for Wigner-type matrices [arXiv:1809.03971,arXiv:1811.04055]. We also extend the finite dimensional band mass formula from [arXiv:1804.07744] to the von Neumann algebra setting by showing that the spectral mass of the bands is topologically rigid under deformations and we conclude that these masses are quantized in some important cases.}, author = {Alt, Johannes and Erdös, László and Krüger, Torben H}, booktitle = {arXiv}, title = {{The Dyson equation with linear self-energy: Spectral bands, edges and cusps}}, year = {2018}, } @phdthesis{149, abstract = {The eigenvalue density of many large random matrices is well approximated by a deterministic measure, the self-consistent density of states. In the present work, we show this behaviour for several classes of random matrices. In fact, we establish that, in each of these classes, the self-consistent density of states approximates the eigenvalue density of the random matrix on all scales slightly above the typical eigenvalue spacing. For large classes of random matrices, the self-consistent density of states exhibits several universal features. We prove that, under suitable assumptions, random Gram matrices and Hermitian random matrices with decaying correlations have a 1/3-Hölder continuous self-consistent density of states ρ on R, which is analytic, where it is positive, and has either a square root edge or a cubic root cusp, where it vanishes. We, thus, extend the validity of the corresponding result for Wigner-type matrices from [4, 5, 7]. We show that ρ is determined as the inverse Stieltjes transform of the normalized trace of the unique solution m(z) to the Dyson equation −m(z) −1 = z − a + S[m(z)] on C N×N with the constraint Im m(z) ≥ 0. Here, z lies in the complex upper half-plane, a is a self-adjoint element of C N×N and S is a positivity-preserving operator on C N×N encoding the first two moments of the random matrix. In order to analyze a possible limit of ρ for N → ∞ and address some applications in free probability theory, we also consider the Dyson equation on infinite dimensional von Neumann algebras. We present two applications to random matrices. We first establish that, under certain assumptions, large random matrices with independent entries have a rotationally symmetric self-consistent density of states which is supported on a centered disk in C. Moreover, it is infinitely often differentiable apart from a jump on the boundary of this disk. Second, we show edge universality at all regular (not necessarily extreme) spectral edges for Hermitian random matrices with decaying correlations.}, author = {Alt, Johannes}, issn = {2663-337X}, pages = {456}, publisher = {Institute of Science and Technology Austria}, title = {{Dyson equation and eigenvalue statistics of random matrices}}, doi = {10.15479/AT:ISTA:TH_1040}, year = {2018}, }