@misc{2765,
abstract = {This is a study of the universality of spectral statistics for large random matrices. Considered are N×N symmetric, Hermitian, or quaternion self-dual random matrices with independent identically distributed entries (Wigner matrices), where the probability distribution of each matrix element is given by a measure v with zero expectation and with subexponential decay. The main result is that the correlation functions of the local eigenvalue statistics in the bulk of the spectrum coincide with those of the Gaussian Orthogonal Ensemble (GOE), the Gaussian Unitary Ensemble (GUE), and the Gaussian Symplectic Ensemble (GSE), respectively, in the limit as N → ∞. This approach is based on a study of the Dyson Brownian motion via a related new dynamics, the local relaxation flow. As a main input, it is established that the density of the eigenvalues converges to the Wigner semicircle law, and this holds even down to the smallest possible scale. Moreover, it is shown that the eigenvectors are completely delocalized. These results hold even without the condition that the matrix elements are identically distributed: only independence is used. In fact, for the matrix elements of the Green function strong estimates are given that imply that the local statistics of any two ensembles in the bulk are identical if the first four moments of the matrix elements match. Universality at the spectral edges requires matching only two moments. A Wigner-type estimate is also proved, and it is shown that the eigenvalues repel each other on arbitrarily small scales.},
author = {László Erdös},
booktitle = {Russian Mathematical Surveys},
number = {3},
pages = {507 -- 626},
publisher = {IOP Publishing Ltd.},
title = {{Universality of Wigner random matrices: A survey of recent results}},
doi = {10.1070/RM2011v066n03ABEH004749},
volume = {66},
year = {2011},
}
@article{2766,
abstract = {We consider Hermitian and symmetric random band matrices H in d ≥ dimensions. The matrix elements Hxy, indexed by x,y ∈ Λ ⊂ ℤd are independent and their variances satisfy σ2xy:= E{pipe}Hxy{pipe}2 = W-d f((x-y)/W for some probability density f. We assume that the law of each matrix element Hxy is symmetric and exhibits subexponential decay. We prove that the time evolution of a quantum particle subject to the Hamiltonian H is diffusive on time scales ≪ Wd/3. We also show that the localization length of the eigenvectors of H is larger than a factor Wd/6 times the band width W. All results are uniform in the size {pipe}Λ{pipe} of the matrix. This extends our recent result (Erdo{double acute}s and Knowles in Commun. Math. Phys., 2011) to general band matrices. As another consequence of our proof we show that, for a larger class of random matrices satisfying Σx σ2xy for all y, the largest eigenvalue of H is bounded with high probability by 2+M-2/3+e{open} for any e{open} > 0, where M:= 1/(maxx,y σ2xy).},
author = {László Erdös and Knowles, Antti},
journal = {Annales Henri Poincare},
number = {7},
pages = {1227 -- 1319},
publisher = {Birkhäuser},
title = {{Quantum diffusion and delocalization for band matrices with general distribution}},
doi = {10.1007/s00023-011-0104-5},
volume = {12},
year = {2011},
}
@article{2799,
abstract = {Shear flows undergo a sudden transition from laminar to turbulent motion as the velocity increases, and the onset of turbulence radically changes transport efficiency and mixing properties. Even for the well-studied case of pipe flow, it has not been possible to determine at what Reynolds number the motion will be either persistently turbulent or ultimately laminar. We show that in pipes, turbulence that is transient at low Reynolds numbers becomes sustained at a distinct critical point. Through extensive experiments and computer simulations, we were able to identify and characterize the processes ultimately responsible for sustaining turbulence. In contrast to the classical Landau-Ruelle-Takens view that turbulence arises from an increase in the temporal complexity of fluid motion, here, spatial proliferation of chaotic domains is the decisive process and intrinsic to the nature of fluid turbulence.},
author = {Avila, Kerstin and Moxey, David and de Lózar, Alberto and Avila, Marc and Barkley, Dwight and Björn Hof},
journal = {Science},
number = {6039},
pages = {192 -- 196},
publisher = {American Association for the Advancement of Science},
title = {{The onset of turbulence in pipe flow}},
doi = {10.1126/science.1203223},
volume = {333},
year = {2011},
}
@article{2800,
abstract = {In shear flows, turbulence first occurs in the form of localized structures (puffs/spots) surrounded by laminar fluid. We here investigate such spatially intermittent flows in a pipe experiment showing that turbulent puffs have a well-defined interaction distance, which sets their minimum spacing as well as the maximum observable turbulent fraction. Two methodologies are employed. Starting from a laminar flow, puffs are first created by locally injecting a jet of fluid through the pipe wall. When the perturbation is applied periodically at low frequencies, as expected, a regular sequence of puffs is observed where the puff spacing is given by the ratio of the mean flow speed to the perturbation frequency. At large frequencies however puffs are found to interact and annihilate each other. Varying the perturbation frequency, an interaction distance is determined which sets the highest possible turbulence fraction. This enables us to establish an upper bound for the friction factor in the transitional regime, which provides a well-defined link between the Blasius and the Hagen-Poiseuille friction laws. In the second set of experiments, the Reynolds number is reduced suddenly from fully turbulent to the intermittent regime. The resulting flow reorganizes itself to a sequence of constant size puffs which, unlike in Couette and Taylor–Couette flow are randomly spaced. The minimum distance between the turbulent patches is identical to the puff interaction length. The puff interaction length is found to be in agreement with the wavelength of regular stripe and spiral patterns in plane Couette and Taylor–Couette flow.},
author = {Samanta, Devranjan and de Lózar, Alberto and Björn Hof},
journal = {Journal of Fluid Mechanics},
pages = {193 -- 204},
publisher = {Cambridge University Press},
title = {{Experimental investigation of laminar turbulent intermittency in pipe flow}},
doi = {10.1017/jfm.2011.189},
volume = {681},
year = {2011},
}
@inproceedings{2801,
abstract = {Turbulent puffs in pipe flow are characterized by a sharp laminar-turbulent interface at the trailing edge and a more diffused leading interface. It is known that these laminar-turbulent interfaces propagate at a speed that is approximately equal to the flow rate. Our results from direct numerical simulation show that, locally, the interface velocity relative to the fluid (i) counteracts the advection due to the laminar velocity profile so that the puff can preserve its characteristic overall shape, (ii) is very small in magnitude, but involves a large interface area so that the global propagation velocity relative to the mean flow can be large and (iii) is determined by both inertial and viscous effects. The analysis provides some new insights into the mechanisms that sustain or expand localized turbulence and might be relevant for the design of new control strategies.},
author = {Holzner, Markus and Avila, Marc and de Lózar, Alberto and Björn Hof},
number = {5},
publisher = {IOP Publishing Ltd.},
title = {{A Lagrangian approach to the interface velocity of turbulent puffs in pipe flow}},
doi = {10.1088/1742-6596/318/5/052031},
volume = {318},
year = {2011},
}