@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{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}, } @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{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{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{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{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}, booktitle = {Proceedings of the International Congress of Mathematicians}, location = {Seoul, Korea}, pages = {214 -- 236}, publisher = {International Congress of Mathematicians}, title = {{Random matrices, log-gases and Hölder regularity}}, volume = {3}, 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 = {1083-6489}, 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{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}, }