@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{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{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{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},
}
@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{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{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{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},
}
@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{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},
}
@article{1824,
abstract = {Condensation phenomena arise through a collective behaviour of particles. They are observed in both classical and quantum systems, ranging from the formation of traffic jams in mass transport models to the macroscopic occupation of the energetic ground state in ultra-cold bosonic gases (Bose-Einstein condensation). Recently, it has been shown that a driven and dissipative system of bosons may form multiple condensates. Which states become the condensates has, however, remained elusive thus far. The dynamics of this condensation are described by coupled birth-death processes, which also occur in evolutionary game theory. Here we apply concepts from evolutionary game theory to explain the formation of multiple condensates in such driven-dissipative bosonic systems. We show that the vanishing of relative entropy production determines their selection. The condensation proceeds exponentially fast, but the system never comes to rest. Instead, the occupation numbers of condensates may oscillate, as we demonstrate for a rock-paper-scissors game of condensates.},
author = {Knebel, Johannes and Weber, Markus and Krüger, Torben H and Frey, Erwin},
journal = {Nature Communications},
publisher = {Nature Publishing Group},
title = {{Evolutionary games of condensates in coupled birth-death processes}},
doi = {10.1038/ncomms7977},
volume = {6},
year = {2015},
}
@article{1864,
abstract = {The Altshuler–Shklovskii formulas (Altshuler and Shklovskii, BZh Eksp Teor Fiz 91:200, 1986) predict, for any disordered quantum system in the diffusive regime, a universal power law behaviour for the correlation functions of the mesoscopic eigenvalue density. In this paper and its companion (Erdős and Knowles, The Altshuler–Shklovskii formulas for random band matrices I: the unimodular case, 2013), we prove these formulas for random band matrices. In (Erdős and Knowles, The Altshuler–Shklovskii formulas for random band matrices I: the unimodular case, 2013) we introduced a diagrammatic approach and presented robust estimates on general diagrams under certain simplifying assumptions. In this paper, we remove these assumptions by giving a general estimate of the subleading diagrams. We also give a precise analysis of the leading diagrams which give rise to the Altschuler–Shklovskii power laws. Moreover, we introduce a family of general random band matrices which interpolates between real symmetric (β = 1) and complex Hermitian (β = 2) models, and track the transition for the mesoscopic density–density correlation. Finally, we address the higher-order correlation functions by proving that they behave asymptotically according to a Gaussian process whose covariance is given by the Altshuler–Shklovskii formulas.
},
author = {Erdös, László and Knowles, Antti},
journal = {Annales Henri Poincare},
number = {3},
pages = {709 -- 799},
publisher = {Springer},
title = {{The Altshuler–Shklovskii formulas for random band matrices II: The general case}},
doi = {10.1007/s00023-014-0333-5},
volume = {16},
year = {2015},
}
@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{2166,
abstract = {We consider the spectral statistics of large random band matrices on mesoscopic energy scales. We show that the correlation function of the local eigenvalue density exhibits a universal power law behaviour that differs from the Wigner-Dyson- Mehta statistics. This law had been predicted in the physics literature by Altshuler and Shklovskii in (Zh Eksp Teor Fiz (Sov Phys JETP) 91(64):220(127), 1986); it describes the correlations of the eigenvalue density in general metallic sampleswith weak disorder. Our result rigorously establishes the Altshuler-Shklovskii formulas for band matrices. In two dimensions, where the leading term vanishes owing to an algebraic cancellation, we identify the first non-vanishing term and show that it differs substantially from the prediction of Kravtsov and Lerner in (Phys Rev Lett 74:2563-2566, 1995). The proof is given in the current paper and its companion (Ann. H. Poincaré. arXiv:1309.5107, 2014). },
author = {Erdös, László and Knowles, Antti},
journal = {Communications in Mathematical Physics},
number = {3},
pages = {1365 -- 1416},
publisher = {Springer},
title = {{The Altshuler-Shklovskii formulas for random band matrices I: the unimodular case}},
doi = {10.1007/s00220-014-2119-5},
volume = {333},
year = {2015},
}
@article{1926,
abstract = {We consider cross products of finite graphs with a class of trees that have arbitrarily but finitely long line segments, such as the Fibonacci tree. Such cross products are called tree-strips. We prove that for small disorder random Schrödinger operators on such tree-strips have purely absolutely continuous spectrum in a certain set.},
author = {Sadel, Christian},
journal = {Mathematical Physics, Analysis and Geometry},
number = {3-4},
pages = {409 -- 440},
publisher = {Springer},
title = {{Absolutely continuous spectrum for random Schrödinger operators on the Fibonacci and similar Tree-strips}},
doi = {10.1007/s11040-014-9163-4},
volume = {17},
year = {2014},
}
@article{1937,
abstract = {We prove the edge universality of the beta ensembles for any β ≥ 1, provided that the limiting spectrum is supported on a single interval, and the external potential is C4 and regular. We also prove that the edge universality holds for generalized Wigner matrices for all symmetry classes. Moreover, our results allow us to extend bulk universality for beta ensembles from analytic potentials to potentials in class C4.},
author = {Bourgade, Paul and Erdös, László and Yau, Horngtzer},
journal = {Communications in Mathematical Physics},
number = {1},
pages = {261 -- 353},
publisher = {Springer},
title = {{Edge universality of beta ensembles}},
doi = {10.1007/s00220-014-2120-z},
volume = {332},
year = {2014},
}
@inproceedings{1507,
abstract = {The Wigner-Dyson-Gaudin-Mehta conjecture asserts that the local eigenvalue statistics of large real and complex Hermitian matrices with independent, identically distributed entries are universal in a sense that they depend only on the symmetry class of the matrix and otherwise are independent of the details of the distribution. We present the recent solution to this half-century old conjecture. We explain how stochastic tools, such as the Dyson Brownian motion, and PDE ideas, such as De Giorgi-Nash-Moser regularity theory, were combined in the solution. We also show related results for log-gases that represent a universal model for strongly correlated systems. Finally, in the spirit of Wigner’s original vision, we discuss the extensions of these universality results to more realistic physical systems such as random band matrices.},
author = {Erdös, László},
location = {Seoul, Korea},
pages = {214 -- 236},
publisher = {Kyung Moon SA Co. Ltd.},
title = {{Random matrices, log-gases and Hölder regularity}},
volume = {3},
year = {2014},
}
@article{2019,
abstract = {We prove that the empirical density of states of quantum spin glasses on arbitrary graphs converges to a normal distribution as long as the maximal degree is negligible compared with the total number of edges. This extends the recent results of Keating et al. (2014) that were proved for graphs with bounded chromatic number and with symmetric coupling distribution. Furthermore, we generalise the result to arbitrary hypergraphs. We test the optimality of our condition on the maximal degree for p-uniform hypergraphs that correspond to p-spin glass Hamiltonians acting on n distinguishable spin- 1/2 particles. At the critical threshold p = n1/2 we find a sharp classical-quantum phase transition between the normal distribution and the Wigner semicircle law. The former is characteristic to classical systems with commuting variables, while the latter is a signature of noncommutative random matrix theory.},
author = {Erdös, László and Schröder, Dominik J},
journal = {Mathematical Physics, Analysis and Geometry},
number = {3-4},
pages = {441 -- 464},
publisher = {Springer},
title = {{Phase transition in the density of states of quantum spin glasses}},
doi = {10.1007/s11040-014-9164-3},
volume = {17},
year = {2014},
}
@article{2179,
abstract = {We extend the proof of the local semicircle law for generalized Wigner matrices given in MR3068390 to the case when the matrix of variances has an eigenvalue -1. In particular, this result provides a short proof of the optimal local Marchenko-Pastur law at the hard edge (i.e. around zero) for sample covariance matrices X*X, where the variances of the entries of X may vary.},
author = {Ajanki, Oskari H and Erdös, László and Krüger, Torben H},
journal = {Electronic Communications in Probability},
publisher = {Institute of Mathematical Statistics},
title = {{Local semicircle law with imprimitive variance matrix}},
doi = {10.1214/ECP.v19-3121},
volume = {19},
year = {2014},
}
@article{2225,
abstract = {We consider sample covariance matrices of the form X∗X, where X is an M×N matrix with independent random entries. We prove the isotropic local Marchenko-Pastur law, i.e. we prove that the resolvent (X∗X−z)−1 converges to a multiple of the identity in the sense of quadratic forms. More precisely, we establish sharp high-probability bounds on the quantity ⟨v,(X∗X−z)−1w⟩−⟨v,w⟩m(z), where m is the Stieltjes transform of the Marchenko-Pastur law and v,w∈CN. We require the logarithms of the dimensions M and N to be comparable. Our result holds down to scales Iz≥N−1+ε and throughout the entire spectrum away from 0. We also prove analogous results for generalized Wigner matrices.
},
author = {Bloemendal, Alex and Erdös, László and Knowles, Antti and Yau, Horng and Yin, Jun},
issn = {10836489},
journal = {Electronic Journal of Probability},
publisher = {Institute of Mathematical Statistics},
title = {{Isotropic local laws for sample covariance and generalized Wigner matrices}},
doi = {10.1214/EJP.v19-3054},
volume = {19},
year = {2014},
}
@article{2699,
abstract = {We prove the universality of the β-ensembles with convex analytic potentials and for any β >
0, i.e. we show that the spacing distributions of log-gases at any inverse temperature β coincide with those of the Gaussian β-ensembles.},
author = {Erdös, László and Bourgade, Paul and Yau, Horng},
journal = {Duke Mathematical Journal},
number = {6},
pages = {1127 -- 1190},
publisher = {Duke University Press},
title = {{Universality of general β-ensembles}},
doi = {10.1215/00127094-2649752},
volume = {163},
year = {2014},
}
@article{2837,
abstract = {We consider a general class of N × N random matrices whose entries hij are independent up to a symmetry constraint, but not necessarily identically distributed. Our main result is a local semicircle law which improves previous results [17] both in the bulk and at the edge. The error bounds are given in terms of the basic small parameter of the model, maxi,j E|hij|2. As a consequence, we prove the universality of the local n-point correlation functions in the bulk spectrum for a class of matrices whose entries do not have comparable variances, including random band matrices with band width W ≫N1-εn with some εn > 0 and with a negligible mean-field component. In addition, we provide a coherent and pedagogical proof of the local semicircle law, streamlining and strengthening previous arguments from [17, 19, 6].},
author = {Erdös, László and Knowles, Antti and Yau, Horng and Yin, Jun},
journal = {Electronic Journal of Probability},
number = {59},
pages = {1--58},
publisher = {Institute of Mathematical Statistics},
title = {{The local semicircle law for a general class of random matrices}},
doi = {10.1214/EJP.v18-2473},
volume = {18},
year = {2013},
}
@article{2698,
abstract = {We consider non-interacting particles subject to a fixed external potential V and a self-generated magnetic field B. The total energy includes the field energy β∫B2 and we minimize over all particle states and magnetic fields. In the case of spin-1/2 particles this minimization leads to the coupled Maxwell-Pauli system. The parameter β tunes the coupling strength between the field and the particles and it effectively determines the strength of the field. We investigate the stability and the semiclassical asymptotics, h→0, of the total ground state energy E(β,h,V). The relevant parameter measuring the field strength in the semiclassical limit is κ=βh. We are not able to give the exact leading order semiclassical asymptotics uniformly in κ or even for fixed κ. We do however give upper and lower bounds on E with almost matching dependence on κ. In the simultaneous limit h→0 and κ→∞ we show that the standard non-magnetic Weyl asymptotics holds. The same result also holds for the spinless case, i.e. where the Pauli operator is replaced by the Schrödinger operator.},
author = {Erdös, László and Fournais, Søren and Solovej, Jan},
journal = {Journal of the European Mathematical Society},
number = {6},
pages = {2093 -- 2113},
publisher = {European Mathematical Society},
title = {{Stability and semiclassics in self-generated fields}},
doi = {10.4171/JEMS/416},
volume = {15},
year = {2013},
}
@article{2782,
abstract = {We consider random n×n matrices of the form (XX*+YY*)^{-1/2}YY*(XX*+YY*)^{-1/2}, where X and Y have independent entries with zero mean and variance one. These matrices are the natural generalization of the Gaussian case, which are known as MANOVA matrices and which have joint eigenvalue density given by the third classical ensemble, the Jacobi ensemble. We show that, away from the spectral edge, the eigenvalue density converges to the limiting density of the Jacobi ensemble even on the shortest possible scales of order 1/n (up to log n factors). This result is the analogue of the local Wigner semicircle law and the local Marchenko-Pastur law for general MANOVA matrices.},
author = {Erdös, László and Farrell, Brendan},
journal = {Journal of Statistical Physics},
number = {6},
pages = {1003 -- 1032},
publisher = {Springer},
title = {{Local eigenvalue density for general MANOVA matrices}},
doi = {10.1007/s10955-013-0807-8},
volume = {152},
year = {2013},
}