@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{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{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 rΔdr+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},
}
@article{1881,
abstract = {We consider random matrices of the form H=W+λV, λ∈ℝ+, where W is a real symmetric or complex Hermitian Wigner matrix of size N and V is a real bounded diagonal random matrix of size N with i.i.d.\ entries that are independent of W. We assume subexponential decay for the matrix entries of W and we choose λ∼1, so that the eigenvalues of W and λV are typically of the same order. Further, we assume that the density of the entries of V is supported on a single interval and is convex near the edges of its support. In this paper we prove that there is λ+∈ℝ+ such that the largest eigenvalues of H are in the limit of large N determined by the order statistics of V for λ>λ+. In particular, the largest eigenvalue of H has a Weibull distribution in the limit N→∞ if λ>λ+. Moreover, for N sufficiently large, we show that the eigenvectors associated to the largest eigenvalues are partially localized for λ>λ+, while they are completely delocalized for λ<λ+. Similar results hold for the lowest eigenvalues. },
author = {Lee, Jioon and Schnelli, Kevin},
journal = {Probability Theory and Related Fields},
number = {1-2},
pages = {165 -- 241},
publisher = {Springer},
title = {{Extremal eigenvalues and eigenvectors of deformed Wigner matrices}},
doi = {10.1007/s00440-014-0610-8},
volume = {164},
year = {2016},
}
@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},
}