@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 = {9781470436483},
pages = {226},
publisher = {American Mathematical Society},
title = {{A dynamical approach to random matrix theory}},
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{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{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{1157,
abstract = {We consider sample covariance matrices of the form Q = ( σ1/2X)(σ1/2X)∗, where the sample X is an M ×N random matrix whose entries are real independent random variables with variance 1/N and whereσ is an M × M positive-definite deterministic matrix. We analyze the asymptotic fluctuations of the largest rescaled eigenvalue of Q when both M and N tend to infinity with N/M →d ϵ (0,∞). For a large class of populations σ in the sub-critical regime, we show that the distribution of the largest rescaled eigenvalue of Q is given by the type-1 Tracy-Widom distribution under the additional assumptions that (1) either the entries of X are i.i.d. Gaussians or (2) that σ is diagonal and that the entries of X have a sub-exponential decay.},
author = {Lee, Ji and Schnelli, Kevin},
journal = {Annals of Applied Probability},
number = {6},
pages = {3786 -- 3839},
publisher = {Institute of Mathematical Statistics},
title = {{Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population}},
doi = {10.1214/16-AAP1193},
volume = {26},
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},
}
@article{1434,
abstract = {We prove that the system of subordination equations, defining the free additive convolution of two probability measures, is stable away from the edges of the support and blow-up singularities by showing that the recent smoothness condition of Kargin is always satisfied. As an application, we consider the local spectral statistics of the random matrix ensemble A+UBU⁎A+UBU⁎, where U is a Haar distributed random unitary or orthogonal matrix, and A and B are deterministic matrices. In the bulk regime, we prove that the empirical spectral distribution of A+UBU⁎A+UBU⁎ concentrates around the free additive convolution of the spectral distributions of A and B on scales down to N−2/3N−2/3.},
author = {Bao, Zhigang and Erdös, László and Schnelli, Kevin},
journal = {Journal of Functional Analysis},
number = {3},
pages = {672 -- 719},
publisher = {Academic Press},
title = {{Local stability of the free additive convolution}},
doi = {10.1016/j.jfa.2016.04.006},
volume = {271},
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 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{1505,
abstract = {This paper is aimed at deriving the universality of the largest eigenvalue of a class of high-dimensional real or complex sample covariance matrices of the form W N =Σ 1/2XX∗Σ 1/2 . Here, X = (xij )M,N is an M× N random matrix with independent entries xij , 1 ≤ i M,≤ 1 ≤ j ≤ N such that Exij = 0, E|xij |2 = 1/N . On dimensionality, we assume that M = M(N) and N/M → d ε (0, ∞) as N ∞→. For a class of general deterministic positive-definite M × M matrices Σ , under some additional assumptions on the distribution of xij 's, we show that the limiting behavior of the largest eigenvalue of W N is universal, via pursuing a Green function comparison strategy raised in [Probab. Theory Related Fields 154 (2012) 341-407, Adv. Math. 229 (2012) 1435-1515] by Erd″os, Yau and Yin for Wigner matrices and extended by Pillai and Yin [Ann. Appl. Probab. 24 (2014) 935-1001] to sample covariance matrices in the null case (&Epsi = I ). Consequently, in the standard complex case (Ex2 ij = 0), combing this universality property and the results known for Gaussian matrices obtained by El Karoui in [Ann. Probab. 35 (2007) 663-714] (nonsingular case) and Onatski in [Ann. Appl. Probab. 18 (2008) 470-490] (singular case), we show that after an appropriate normalization the largest eigenvalue of W N converges weakly to the type 2 Tracy-Widom distribution TW2 . Moreover, in the real case, we show that whenΣ is spiked with a fixed number of subcritical spikes, the type 1 Tracy-Widom limit TW1 holds for the normalized largest eigenvalue of W N , which extends a result of Féral and Péché in [J. Math. Phys. 50 (2009) 073302] to the scenario of nondiagonal Σ and more generally distributed X . In summary, we establish the Tracy-Widom type universality for the largest eigenvalue of generally distributed sample covariance matrices under quite light assumptions on &Sigma . Applications of these limiting results to statistical signal detection and structure recognition of separable covariance matrices are also discussed.},
author = {Bao, Zhigang and Pan, Guangming and Zhou, Wang},
journal = {Annals of Statistics},
number = {1},
pages = {382 -- 421},
publisher = {Institute of Mathematical Statistics},
title = {{Universality for the largest eigenvalue of sample covariance matrices with general population}},
doi = {10.1214/14-AOS1281},
volume = {43},
year = {2015},
}
@article{1506,
abstract = {Consider the square random matrix An = (aij)n,n, where {aij:= a(n)ij , i, j = 1, . . . , n} is a collection of independent real random variables with means zero and variances one. Under the additional moment condition supn max1≤i,j ≤n Ea4ij <∞, we prove Girko's logarithmic law of det An in the sense that as n→∞ log | detAn| ? (1/2) log(n-1)! d/→√(1/2) log n N(0, 1).},
author = {Bao, Zhigang and Pan, Guangming and Zhou, Wang},
journal = {Bernoulli},
number = {3},
pages = {1600 -- 1628},
publisher = {Bernoulli Society for Mathematical Statistics and Probability},
title = {{The logarithmic law of random determinant}},
doi = {10.3150/14-BEJ615},
volume = {21},
year = {2015},
}
@article{1508,
abstract = {We consider generalized Wigner ensembles and general β-ensembles with analytic potentials for any β ≥ 1. The recent universality results in particular assert that the local averages of consecutive eigenvalue gaps in the bulk of the spectrum are universal in the sense that they coincide with those of the corresponding Gaussian β-ensembles. In this article, we show that local averaging is not necessary for this result, i.e. we prove that the single gap distributions in the bulk are universal. In fact, with an additional step, our result can be extended to any C4(ℝ) potential.},
author = {Erdös, László and Yau, Horng},
journal = {Journal of the European Mathematical Society},
number = {8},
pages = {1927 -- 2036},
publisher = {European Mathematical Society},
title = {{Gap universality of generalized Wigner and β ensembles}},
doi = {10.4171/JEMS/548},
volume = {17},
year = {2015},
}
@article{1585,
abstract = {In this paper, we consider the fluctuation of mutual information statistics of a multiple input multiple output channel communication systems without assuming that the entries of the channel matrix have zero pseudovariance. To this end, we also establish a central limit theorem of the linear spectral statistics for sample covariance matrices under general moment conditions by removing the restrictions imposed on the second moment and fourth moment on the matrix entries in Bai and Silverstein (2004).},
author = {Bao, Zhigang and Pan, Guangming and Zhou, Wang},
journal = {IEEE Transactions on Information Theory},
number = {6},
pages = {3413 -- 3426},
publisher = {IEEE},
title = {{Asymptotic mutual information statistics of MIMO channels and CLT of sample covariance matrices}},
doi = {10.1109/TIT.2015.2421894},
volume = {61},
year = {2015},
}
@article{1674,
abstract = {We consider N × N random matrices of the form H = W + V where W is a real symmetric Wigner matrix and V 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 of W and V are typically of the same order. For a large class of diagonal matrices V, we show that the rescaled distribution of the extremal eigenvalues is given by the Tracy-Widom distribution F1 in the limit of large N. Our proofs also apply to the complex Hermitian setting, i.e. when W is a complex Hermitian Wigner matrix.},
author = {Lee, Jioon and Schnelli, Kevin},
journal = {Reviews in Mathematical Physics},
number = {8},
publisher = {World Scientific Publishing},
title = {{Edge universality for deformed Wigner matrices}},
doi = {10.1142/S0129055X1550018X},
volume = {27},
year = {2015},
}
@article{1677,
abstract = {We consider real symmetric and complex Hermitian random matrices with the additional symmetry hxy = hN-y,N-x. The matrix elements are independent (up to the fourfold symmetry) and not necessarily identically distributed. This ensemble naturally arises as the Fourier transform of a Gaussian orthogonal ensemble. Italso occurs as the flip matrix model - an approximation of the two-dimensional Anderson model at small disorder. We show that the density of states converges to the Wigner semicircle law despite the new symmetry type. We also prove the local version of the semicircle law on the optimal scale.},
author = {Alt, Johannes},
journal = {Journal of Mathematical Physics},
number = {10},
publisher = {American Institute of Physics},
title = {{The local semicircle law for random matrices with a fourfold symmetry}},
doi = {10.1063/1.4932606},
volume = {56},
year = {2015},
}