Spectral fluctuations of tridiagonal random matrices from the beta-Hermite ensemble

A time series delta(n), the fluctuation of the nth unfolded eigenvalue was recently characterized for the classical Gaussian ensembles of NxN random matrices (GOE, GUE, GSE). It is investigated here for the beta-Hermite ensemble as a function of beta (zero or positive) by Monte Carlo simulations. The fluctuation of delta(n) and the autocorrelation function vary logarithmically with n for any beta>0 (1<<n<<N). The simple logarithmic behavior reported for the higher-order moments of delta(n) for the GOE (beta=1) and the GUE (beta=2) is valid for any positive beta and is accounted for by Gaussian distributions whose variances depend linearly on ln(n). The 1/f noise previously demonstrated for delta(n) series of the three Gaussian ensembles, is characterized by wavelet analysis both as a function of beta and of N. When beta decreases from 1 to 0, for a given and large enough N, the evolution from a 1/f noise at beta=1 to a 1/f^2 noise at beta=0 is heterogeneous with a ~1/f^2 noise at the finest scales and a ~1/f noise at the coarsest ones. The range of scales in which a ~1/f^2 noise predominates grows progressively when beta decreases. Asymptotically, a 1/f^2 noise is found for beta=0 while a 1/f noise is the rule for beta positive.Comment: 35 pages, 10 figures, corresponding author: G. Le Cae

Differences between regular and random order of updates in damage spreading simulations

We investigate the spreading of damage in the three-dimensional Ising model by means of large-scale Monte-Carlo simulations. Within the Glauber dynamics we use different rules for the order in which the sites are updated. We find that the stationary damage values and the spreading temperature are different for different update order. In particular, random update order leads to larger damage and a lower spreading temperature than regular order. Consequently, damage spreading in the Ising model is non-universal not only with respect to different update algorithms (e.g. Glauber vs. heat-bath dynamics) as already known, but even with respect to the order of sites.Comment: final version as published, 4 pages REVTeX, 2 eps figures include

Chaotic behavior and damage spreading in the Glauber Ising model - a master equation approach

We investigate the sensitivity of the time evolution of a kinetic Ising model with Glauber dynamics against the initial conditions. To do so we apply the "damage spreading" method, i.e., we study the simultaneous evolution of two identical systems subjected to the same thermal noise. We derive a master equation for the joint probability distribution of the two systems. We then solve this master equation within an effective-field approximation which goes beyond the usual mean-field approximation by retaining the fluctuations though in a quite simplistic manner. The resulting effective-field theory is applied to different physical situations. It is used to analyze the fixed points of the master equation and their stability and to identify regular and chaotic phases of the Glauber Ising model. We also discuss the relation of our results to directed percolation.Comment: 9 pages RevTeX, 4 EPS figure

New Monte Carlo method for planar Poisson-Voronoi cells

By a new Monte Carlo algorithm we evaluate the sidedness probability p_n of a planar Poisson-Voronoi cell in the range 3 \leq n \leq 1600. The algorithm is developed on the basis of earlier theoretical work; it exploits, in particular, the known asymptotic behavior of p_n as n\to\infty. Our p_n values all have between four and six significant digits. Accurate n dependent averages, second moments, and variances are obtained for the cell area and the cell perimeter. The numerical large n behavior of these quantities is analyzed in terms of asymptotic power series in 1/n. Snapshots are shown of typical occurrences of extremely rare events implicating cells of up to n=1600 sides embedded in an ordinary Poisson-Voronoi diagram. We reveal and discuss the characteristic features of such many-sided cells and their immediate environment. Their relevance for observable properties is stressed.Comment: 35 pages including 10 figures and 4 table

Experimental evidence of ageing and slow restoration of the weak-contact configuration in tilted 3D granular packings

Granular packings slowly driven towards their instability threshold are studied using a digital imaging technique as well as a nonlinear acoustic method. The former method allows us to study grain rearrangements on the surface during the tilting and the latter enables to selectively probe the modifications of the weak-contact fraction in the material bulk. Gradual ageing of both the surface activity and the weak-contact reconfigurations is observed as a result of repeated tilt cycles up to a given angle smaller than the angle of avalanche. For an aged configuration reached after several consecutive tilt cycles, abrupt resumption of the on-surface activity and of the weak-contact rearrangements occurs when the packing is subsequently inclined beyond the previous maximal tilting angle. This behavior is compared with literature results from numerical simulations of inclined 2D packings. It is also found that the aged weak-contact configurations exhibit spontaneous restoration towards the initial state if the packing remains at rest for tens of minutes. When the packing is titled forth and back between zero and near-critical angles, instead of ageing, the weak-contact configuration exhibits "internal weak-contact avalanches" in the vicinity of both the near-critical and zero angles. By contrast, the stronger-contact skeleton remains stable

The perimeter of large planar Voronoi cells: a double-stranded random walk

Let $p\_n$ be the probability for a planar Poisson-Voronoi cell to have exactly $n$ sides. We construct the asymptotic expansion of $\log p\_n$ up to terms that vanish as $n\to\infty$. We show that {\it two independent biased random walks} executed by the polar angle determine the trajectory of the cell perimeter. We find the limit distribution of (i) the angle between two successive vertex vectors, and (ii) the one between two successive perimeter segments. We obtain the probability law for the perimeter's long wavelength deviations from circularity. We prove Lewis' law and show that it has coefficient 1/4.Comment: Slightly extended version; journal reference adde

Damage spreading in random field systems

We investigate how a quenched random field influences the damage spreading transition in kinetic Ising models. To this end we generalize a recent master equation approach and derive an effective field theory for damage spreading in random field systems. This theory is applied to the Glauber Ising model with a bimodal random field distribution. We find that the random field influences the spreading transition by two different mechanisms with opposite effects. First, the random field favors the same particular direction of the spin variable at each site in both systems which reduces the damage. Second, the random field suppresses the magnetization which, in turn, tends to increase the damage. The competition between these two effects leads to a rich behavior.Comment: 4 pages RevTeX, 3 eps figure

Asymptotic statistics of the n-sided planar Poisson-Voronoi cell. I. Exact results

We achieve a detailed understanding of the $n$-sided planar Poisson-Voronoi cell in the limit of large $n$. Let ${p}\_n$ be the probability for a cell to have $n$ sides. We construct the asymptotic expansion of $\log {p}\_n$ up to terms that vanish as $n\to\infty$. We obtain the statistics of the lengths of the perimeter segments and of the angles between adjoining segments: to leading order as $n\to\infty$, and after appropriate scaling, these become independent random variables whose laws we determine; and to next order in $1/n$ they have nontrivial long range correlations whose expressions we provide. The $n$-sided cell tends towards a circle of radius (n/4\pi\lambda)^{\half}, where $\lambda$ is the cell density; hence Lewis' law for the average area $A\_n$ of the $n$-sided cell behaves as $A\_n \simeq cn/\lambda$ with $c=1/4$. For $n\to\infty$ the cell perimeter, expressed as a function $R(\phi)$ of the polar angle $\phi$, satisfies $d^2 R/d\phi^2 = F(\phi)$, where $F$ is known Gaussian noise; we deduce from it the probability law for the perimeter's long wavelength deviations from circularity. Many other quantities related to the asymptotic cell shape become accessible to calculation.Comment: 54 pages, 3 figure

Structure et dynamique de la liaison hydrogène dans l'eau confinée ou aux interfaces

International audienceLorsque de l'eau est confinée dans une cavité nanométrique, ses propriétés structurales et dynamiques sont modifiées par rapport à celles de l'eau dans le volume. Les propriétés de l'eau confinée dans des oxydes présentent un intérêt à la fois fondamental et pratique, mais l'état du réseau percolatif dans ces cavités rigides est mal connu. Jusqu'à présent, la majorité des résultats a été obtenue sur de l'eau confinée dans de la matière molle. Nous présenterons ici des résultats obtenus sur la structure et la dynamique de l'eau confinée dans des systèmes rigides comme les oxydes. Pour cela, la spectroscopie infrarouge est une technique de choix, que ce soit par la grande gamme spectrale qu'elle offre (de l'infrarouge lointain, qui permet d'étudier des modes collectifs, au moyen infrarouge avec l'étude de la bande d'élongation O-H), mais aussi par la possibilité de faire des études de spectroscopie d'absorption transitoire femtoseconde qui donnent accès à la durée de vie du vibrateur O-H et à la rotation des molécules d'eau. Ces études ont été réalisées sur l'eau de surface (eau sur une surface d'alumine 1 ou à la surface d'un verre de silice 2 ,3) et l'eau confinée dans des géométries particulières, que ce soit un confinement tridimensionnel comme dans les pores de silice nanométrique 2 ou bidimensionnel comme dans les argiles 4. Les différences de comportement dynamique et de structure du réseau de liaisons hydrogène de l'eau confinée dans des systèmes « durs}) comme les oxydes ou « mous}) comme dans les échantillons biologiques (myoglobine concentrées ...) seront également discutées

Compact smallest eigenvalue expressions in Wishart-Laguerre ensembles with or without fixed-trace

The degree of entanglement of random pure states in bipartite quantum systems can be estimated from the distribution of the extreme Schmidt eigenvalues. For a bipartition of size M\geq N, these are distributed according to a Wishart-Laguerre ensemble (WL) of random matrices of size N x M, with a fixed-trace constraint. We first compute the distribution and moments of the smallest eigenvalue in the fixed trace orthogonal WL ensemble for arbitrary M\geq N. Our method is based on a Laplace inversion of the recursive results for the corresponding orthogonal WL ensemble by Edelman. Explicit examples are given for fixed N and M, generalizing and simplifying earlier results. In the microscopic large-N limit with M-N fixed, the orthogonal and unitary WL distributions exhibit universality after a suitable rescaling and are therefore independent of the constraint. We prove that very recent results given in terms of hypergeometric functions of matrix argument are equivalent to more explicit expressions in terms of a Pfaffian or determinant of Bessel functions. While the latter were mostly known from the random matrix literature on the QCD Dirac operator spectrum, we also derive some new results in the orthogonal symmetry class.Comment: 25 pag., 4 fig - minor changes, typos fixed. To appear in JSTA