@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 = {EMS Press}, title = {{Gap universality of generalized Wigner and β ensembles}}, doi = {10.4171/JEMS/548}, volume = {17}, 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 = {AIP Publishing}, title = {{The local semicircle law for random matrices with a fourfold symmetry}}, doi = {10.1063/1.4932606}, volume = {56}, year = {2015}, } @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{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{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}, } @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{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}, } @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{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}, } @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 = {EMS Press}, title = {{Stability and semiclassics in self-generated fields}}, doi = {10.4171/JEMS/416}, volume = {15}, year = {2013}, }