@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{721, abstract = {Let S be a positivity-preserving symmetric linear operator acting on bounded functions. The nonlinear equation -1/m=z+Sm with a parameter z in the complex upper half-plane ℍ has a unique solution m with values in ℍ. We show that the z-dependence of this solution can be represented as the Stieltjes transforms of a family of probability measures v on ℝ. Under suitable conditions on S, we show that v has a real analytic density apart from finitely many algebraic singularities of degree at most 3. Our motivation comes from large random matrices. The solution m determines the density of eigenvalues of two prominent matrix ensembles: (i) matrices with centered independent entries whose variances are given by S and (ii) matrices with correlated entries with a translation-invariant correlation structure. Our analysis shows that the limiting eigenvalue density has only square root singularities or cubic root cusps; no other singularities occur.}, author = {Ajanki, Oskari H and Krüger, Torben H and Erdös, László}, issn = {00103640}, journal = {Communications on Pure and Applied Mathematics}, number = {9}, pages = {1672 -- 1705}, publisher = {Wiley-Blackwell}, title = {{Singularities of solutions to quadratic vector equations on the complex upper half plane}}, doi = {10.1002/cpa.21639}, volume = {70}, year = {2017}, } @article{733, abstract = {Let A and B be two N by N deterministic Hermitian matrices and let U be an N by N Haar distributed unitary matrix. It is well known that the spectral distribution of the sum H = A + UBU∗ converges weakly to the free additive convolution of the spectral distributions of A and B, as N tends to infinity. We establish the optimal convergence rate in the bulk of the spectrum.}, author = {Bao, Zhigang and Erdös, László and Schnelli, Kevin}, journal = {Advances in Mathematics}, pages = {251 -- 291}, publisher = {Academic Press}, title = {{Convergence rate for spectral distribution of addition of random matrices}}, doi = {10.1016/j.aim.2017.08.028}, volume = {319}, year = {2017}, } @article{1528, abstract = {We consider N×N Hermitian random matrices H consisting of blocks of size M≥N6/7. The matrix elements are i.i.d. within the blocks, close to a Gaussian in the four moment matching sense, but their distribution varies from block to block to form a block-band structure, with an essential band width M. We show that the entries of the Green’s function G(z)=(H−z)−1 satisfy the local semicircle law with spectral parameter z=E+iη down to the real axis for any η≫N−1, using a combination of the supersymmetry method inspired by Shcherbina (J Stat Phys 155(3): 466–499, 2014) and the Green’s function comparison strategy. Previous estimates were valid only for η≫M−1. The new estimate also implies that the eigenvectors in the middle of the spectrum are fully delocalized.}, author = {Bao, Zhigang and Erdös, László}, issn = {01788051}, journal = {Probability Theory and Related Fields}, number = {3-4}, pages = {673 -- 776}, publisher = {Springer}, title = {{Delocalization for a class of random block band matrices}}, doi = {10.1007/s00440-015-0692-y}, volume = {167}, year = {2017}, } @article{1337, abstract = {We consider the local eigenvalue distribution of large self-adjoint N×N random matrices H=H∗ with centered independent entries. In contrast to previous works the matrix of variances sij=\mathbbmE|hij|2 is not assumed to be stochastic. Hence the density of states is not the Wigner semicircle law. Its possible shapes are described in the companion paper (Ajanki et al. in Quadratic Vector Equations on the Complex Upper Half Plane. arXiv:1506.05095). We show that as N grows, the resolvent, G(z)=(H−z)−1, converges to a diagonal matrix, diag(m(z)), where m(z)=(m1(z),…,mN(z)) solves the vector equation −1/mi(z)=z+∑jsijmj(z) that has been analyzed in Ajanki et al. (Quadratic Vector Equations on the Complex Upper Half Plane. arXiv:1506.05095). We prove a local law down to the smallest spectral resolution scale, and bulk universality for both real symmetric and complex hermitian symmetry classes.}, author = {Ajanki, Oskari H and Erdös, László and Krüger, Torben H}, issn = {01788051}, journal = {Probability Theory and Related Fields}, number = {3-4}, pages = {667 -- 727}, publisher = {Springer}, title = {{Universality for general Wigner-type matrices}}, doi = {10.1007/s00440-016-0740-2}, volume = {169}, year = {2017}, } @article{550, abstract = {For large random matrices X with independent, centered entries but not necessarily identical variances, the eigenvalue density of XX* is well-approximated by a deterministic measure on ℝ. We show that the density of this measure has only square and cubic-root singularities away from zero. We also extend the bulk local law in [5] to the vicinity of these singularities.}, author = {Alt, Johannes}, issn = {1083589X}, journal = {Electronic Communications in Probability}, publisher = {Institute of Mathematical Statistics}, title = {{Singularities of the density of states of random Gram matrices}}, doi = {10.1214/17-ECP97}, volume = {22}, year = {2017}, } @book{567, abstract = {This book is a concise and self-contained introduction of recent techniques to prove local spectral universality for large random matrices. Random matrix theory is a fast expanding research area, and this book mainly focuses on the methods that the authors participated in developing over the past few years. Many other interesting topics are not included, and neither are several new developments within the framework of these methods. The authors have chosen instead to present key concepts that they believe are the core of these methods and should be relevant for future applications. They keep technicalities to a minimum to make the book accessible to graduate students. With this in mind, they include in this book the basic notions and tools for high-dimensional analysis, such as large deviation, entropy, Dirichlet form, and the logarithmic Sobolev inequality. }, author = {Erdös, László and Yau, Horng}, isbn = {9-781-4704-3648-3}, pages = {226}, publisher = {American Mathematical Society}, title = {{A Dynamical Approach to Random Matrix Theory}}, doi = {10.1090/cln/028}, volume = {28}, year = {2017}, } @article{615, abstract = {We show that the Dyson Brownian Motion exhibits local universality after a very short time assuming that local rigidity and level repulsion of the eigenvalues hold. These conditions are verified, hence bulk spectral universality is proven, for a large class of Wigner-like matrices, including deformed Wigner ensembles and ensembles with non-stochastic variance matrices whose limiting densities differ from Wigner's semicircle law.}, author = {Erdös, László and Schnelli, Kevin}, issn = {02460203}, journal = {Annales de l'institut Henri Poincare (B) Probability and Statistics}, number = {4}, pages = {1606 -- 1656}, publisher = {Institute of Mathematical Statistics}, title = {{Universality for random matrix flows with time dependent density}}, doi = {10.1214/16-AIHP765}, volume = {53}, year = {2017}, } @article{1010, abstract = {We prove a local law in the bulk of the spectrum for random Gram matrices XX∗, a generalization of sample covariance matrices, where X is a large matrix with independent, centered entries with arbitrary variances. The limiting eigenvalue density that generalizes the Marchenko-Pastur law is determined by solving a system of nonlinear equations. Our entrywise and averaged local laws are on the optimal scale with the optimal error bounds. They hold both in the square case (hard edge) and in the properly rectangular case (soft edge). In the latter case we also establish a macroscopic gap away from zero in the spectrum of XX∗. }, author = {Alt, Johannes and Erdös, László and Krüger, Torben H}, issn = {10836489}, journal = {Electronic Journal of Probability}, publisher = {Institute of Mathematical Statistics}, title = {{Local law for random Gram matrices}}, doi = {10.1214/17-EJP42}, volume = {22}, year = {2017}, } @article{1023, abstract = {We consider products of independent square non-Hermitian random matrices. More precisely, let X1,…, Xn be independent N × N random matrices with independent entries (real or complex with independent real and imaginary parts) with zero mean and variance 1/N. Soshnikov-O’Rourke [19] and Götze-Tikhomirov [15] showed that the empirical spectral distribution of the product of n random matrices with iid entries converges to (equation found). We prove that if the entries of the matrices X1,…, Xn are independent (but not necessarily identically distributed) and satisfy uniform subexponential decay condition, then in the bulk the convergence of the ESD of X1,…, Xn to (0.1) holds up to the scale N–1/2+ε.}, author = {Nemish, Yuriy}, issn = {10836489}, journal = {Electronic Journal of Probability}, publisher = {Institute of Mathematical Statistics}, title = {{Local law for the product of independent non-Hermitian random matrices with independent entries}}, doi = {10.1214/17-EJP38}, volume = {22}, year = {2017}, } @article{1144, abstract = {We show that matrix elements of functions of N × N Wigner matrices fluctuate on a scale of order N−1/2 and we identify the limiting fluctuation. Our result holds for any function f of the matrix that has bounded variation thus considerably relaxing the regularity requirement imposed in [7, 11].}, author = {Erdös, László and Schröder, Dominik J}, journal = {Electronic Communications in Probability}, publisher = {Institute of Mathematical Statistics}, title = {{Fluctuations of functions of Wigner matrices}}, doi = {10.1214/16-ECP38}, volume = {21}, year = {2017}, } @article{1207, abstract = {The eigenvalue distribution of the sum of two large Hermitian matrices, when one of them is conjugated by a Haar distributed unitary matrix, is asymptotically given by the free convolution of their spectral distributions. We prove that this convergence also holds locally in the bulk of the spectrum, down to the optimal scales larger than the eigenvalue spacing. The corresponding eigenvectors are fully delocalized. Similar results hold for the sum of two real symmetric matrices, when one is conjugated by Haar orthogonal matrix.}, author = {Bao, Zhigang and Erdös, László and Schnelli, Kevin}, issn = {00103616}, journal = {Communications in Mathematical Physics}, number = {3}, pages = {947 -- 990}, publisher = {Springer}, title = {{Local law of addition of random matrices on optimal scale}}, doi = {10.1007/s00220-016-2805-6}, volume = {349}, year = {2017}, } @article{447, abstract = {We consider last passage percolation (LPP) models with exponentially distributed random variables, which are linked to the totally asymmetric simple exclusion process (TASEP). The competition interface for LPP was introduced and studied in Ferrari and Pimentel (2005a) for cases where the corresponding exclusion process had a rarefaction fan. Here we consider situations with a shock and determine the law of the fluctuations of the competition interface around its deter- ministic law of large number position. We also study the multipoint distribution of the LPP around the shock, extending our one-point result of Ferrari and Nejjar (2015).}, author = {Ferrari, Patrik and Nejjar, Peter}, journal = {Revista Latino-Americana de Probabilidade e Estatística}, pages = {299 -- 325}, publisher = {Instituto Nacional de Matematica Pura e Aplicada}, title = {{Fluctuations of the competition interface in presence of shocks}}, doi = {10.30757/ALEA.v14-17}, volume = {9}, year = {2017}, } @article{483, abstract = {We prove the universality for the eigenvalue gap statistics in the bulk of the spectrum for band matrices, in the regime where the band width is comparable with the dimension of the matrix, W ~ N. All previous results concerning universality of non-Gaussian random matrices are for mean-field models. By relying on a new mean-field reduction technique, we deduce universality from quantum unique ergodicity for band matrices.}, author = {Bourgade, Paul and Erdös, László and Yau, Horng and Yin, Jun}, issn = {10950761}, journal = {Advances in Theoretical and Mathematical Physics}, number = {3}, pages = {739 -- 800}, publisher = {International Press}, title = {{Universality for a class of random band matrices}}, doi = {10.4310/ATMP.2017.v21.n3.a5}, volume = {21}, year = {2017}, } @article{1219, abstract = {We consider N×N random matrices of the form H = W + V where W is a real symmetric or complex Hermitian Wigner matrix and V is a random or deterministic, real, diagonal matrix whose entries are independent of W. We assume subexponential decay for the matrix entries of W, and we choose V so that the eigenvalues ofW and V are typically of the same order. For a large class of diagonal matrices V , we show that the local statistics in the bulk of the spectrum are universal in the limit of large N.}, author = {Lee, Jioon and Schnelli, Kevin and Stetler, Ben and Yau, Horngtzer}, journal = {Annals of Probability}, number = {3}, pages = {2349 -- 2425}, publisher = {Institute of Mathematical Statistics}, title = {{Bulk universality for deformed wigner matrices}}, doi = {10.1214/15-AOP1023}, volume = {44}, year = {2016}, } @article{1223, abstract = {We consider a random Schrödinger operator on the binary tree with a random potential which is the sum of a random radially symmetric potential, Qr, and a random transversally periodic potential, κQt, with coupling constant κ. Using a new one-dimensional dynamical systems approach combined with Jensen's inequality in hyperbolic space (our key estimate) we obtain a fractional moment estimate proving localization for small and large κ. Together with a previous result we therefore obtain a model with two Anderson transitions, from localization to delocalization and back to localization, when increasing κ. As a by-product we also have a partially new proof of one-dimensional Anderson localization at any disorder.}, author = {Froese, Richard and Lee, Darrick and Sadel, Christian and Spitzer, Wolfgang and Stolz, Günter}, journal = {Journal of Spectral Theory}, number = {3}, pages = {557 -- 600}, publisher = {European Mathematical Society}, title = {{Localization for transversally periodic random potentials on binary trees}}, doi = {10.4171/JST/132}, volume = {6}, year = {2016}, } @article{1257, abstract = {We consider products of random matrices that are small, independent identically distributed perturbations of a fixed matrix (Formula presented.). Focusing on the eigenvalues of (Formula presented.) of a particular size we obtain a limit to a SDE in a critical scaling. Previous results required (Formula presented.) to be a (conjugated) unitary matrix so it could not have eigenvalues of different modulus. From the result we can also obtain a limit SDE for the Markov process given by the action of the random products on the flag manifold. Applying the result to random Schrödinger operators we can improve some results by Valko and Virag showing GOE statistics for the rescaled eigenvalue process of a sequence of Anderson models on long boxes. In particular, we solve a problem posed in their work.}, author = {Sadel, Christian and Virág, Bálint}, journal = {Communications in Mathematical Physics}, number = {3}, pages = {881 -- 919}, publisher = {Springer}, title = {{A central limit theorem for products of random matrices and GOE statistics for the Anderson model on long boxes}}, doi = {10.1007/s00220-016-2600-4}, volume = {343}, year = {2016}, } @article{1280, abstract = {We prove the Wigner-Dyson-Mehta conjecture at fixed energy in the bulk of the spectrum for generalized symmetric and Hermitian Wigner matrices. Previous results concerning the universality of random matrices either require an averaging in the energy parameter or they hold only for Hermitian matrices if the energy parameter is fixed. We develop a homogenization theory of the Dyson Brownian motion and show that microscopic universality follows from mesoscopic statistics.}, author = {Bourgade, Paul and Erdös, László and Yau, Horngtzer and Yin, Jun}, journal = {Communications on Pure and Applied Mathematics}, number = {10}, pages = {1815 -- 1881}, publisher = {Wiley-Blackwell}, title = {{Fixed energy universality for generalized wigner matrices}}, doi = {10.1002/cpa.21624}, volume = {69}, year = {2016}, } @article{1489, abstract = {We prove optimal local law, bulk universality and non-trivial decay for the off-diagonal elements of the resolvent for a class of translation invariant Gaussian random matrix ensembles with correlated entries. }, author = {Ajanki, Oskari H and Erdös, László and Krüger, Torben H}, journal = {Journal of Statistical Physics}, number = {2}, pages = {280 -- 302}, publisher = {Springer}, title = {{Local spectral statistics of Gaussian matrices with correlated entries}}, doi = {10.1007/s10955-016-1479-y}, volume = {163}, year = {2016}, } @article{1608, abstract = {We show that the Anderson model has a transition from localization to delocalization at exactly 2 dimensional growth rate on antitrees with normalized edge weights which are certain discrete graphs. The kinetic part has a one-dimensional structure allowing a description through transfer matrices which involve some Schur complement. For such operators we introduce the notion of having one propagating channel and extend theorems from the theory of one-dimensional Jacobi operators that relate the behavior of transfer matrices with the spectrum. These theorems are then applied to the considered model. In essence, in a certain energy region the kinetic part averages the random potentials along shells and the transfer matrices behave similar as for a one-dimensional operator with random potential of decaying variance. At d dimensional growth for d>2 this effective decay is strong enough to obtain absolutely continuous spectrum, whereas for some uniform d dimensional growth with d<2 one has pure point spectrum in this energy region. At exactly uniform 2 dimensional growth also some singular continuous spectrum appears, at least at small disorder. As a corollary we also obtain a change from singular spectrum (d≤2) to absolutely continuous spectrum (d≥3) for random operators of the type rΔdr+λ on ℤd, where r is an orthogonal radial projection, Δd the discrete adjacency operator (Laplacian) on ℤd and λ a random potential. }, author = {Sadel, Christian}, journal = {Annales Henri Poincare}, number = {7}, pages = {1631 -- 1675}, publisher = {Birkhäuser}, title = {{Anderson transition at 2 dimensional growth rate on antitrees and spectral theory for operators with one propagating channel}}, doi = {10.1007/s00023-015-0456-3}, volume = {17}, year = {2016}, }