@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 = {0010-3640}, journal = {Communications on Pure and Applied Mathematics}, number = {9}, pages = {1672 -- 1705}, publisher = {Wiley}, 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{483, abstract = {We prove the universality for the eigenvalue gap statistics in the bulk of the spectrum for band matrices, in the regime where the band width is comparable with the dimension of the matrix, W ~ N. All previous results concerning universality of non-Gaussian random matrices are for mean-field models. By relying on a new mean-field reduction technique, we deduce universality from quantum unique ergodicity for band matrices.}, author = {Bourgade, Paul and Erdös, László and Yau, Horng and Yin, Jun}, issn = {1095-0761}, journal = {Advances in Theoretical and Mathematical Physics}, number = {3}, pages = {739 -- 800}, publisher = {International Press of Boston}, title = {{Universality for a class of random band matrices}}, doi = {10.4310/ATMP.2017.v21.n3.a5}, volume = {21}, 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{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{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 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{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{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 = {EMS Press}, title = {{Localization for transversally periodic random potentials on binary trees}}, doi = {10.4171/JST/132}, volume = {6}, year = {2016}, } @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{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{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{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{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}, }