Factorization of correlations in two-dimensional percolation on the plane and torus

Recently, Delfino and Viti have examined the factorization of the three-point density correlation function P_3 at the percolation point in terms of the two-point density correlation functions P_2. According to conformal invariance, this factorization is exact on the infinite plane, such that the ratio R(z_1, z_2, z_3) = P_3(z_1, z_2, z_3) [P_2(z_1, z_2) P_2(z_1, z_3) P_2(z_2, z_3)]^{1/2} is not only universal but also a constant, independent of the z_i, and in fact an operator product expansion (OPE) coefficient. Delfino and Viti analytically calculate its value (1.022013...) for percolation, in agreement with the numerical value 1.022 found previously in a study of R on the conformally equivalent cylinder. In this paper we confirm the factorization on the plane numerically using periodic lattices (tori) of very large size, which locally approximate a plane. We also investigate the general behavior of R on the torus, and find a minimum value of R approx. 1.0132 when the three points are maximally separated. In addition, we present a simplified expression for R on the plane as a function of the SLE parameter kappa.Comment: Small corrections (final version). In press, J. Phys.

Une ambiance lumineuse en fonction des besoins

Presentation of the ECAL-EPFL collaborative project in user-robot swarm interaction

Unstable decay and state selection II

The decay of unstable states when several metastable states are available for occupation is investigated using path-integral techniques. Specifically, a method is described which allows the probabilities with which the metastable states are occupied to be calculated by finding optimal paths, and fluctuations about them, in the weak noise limit. The method is illustrated on a system described by two coupled Langevin equations, which are found in the study of instabilities in fluid dynamics and superconductivity. The problem involves a subtle interplay between non-linearities and noise, and a naive approximation scheme which does not take this into account is shown to be unsatisfactory. The use of optimal paths is briefly reviewed and then applied to finding the conditional probability of ending up in one of the metastable states, having begun in the unstable state. There are several aspects of the calculation which distinguish it from most others involving optimal paths: (i) the paths do not begin and end on an attractor, and moreover, the final point is to a large extent arbitrary, (ii) the interplay between the fluctuations and the leading order contribution are at the heart of the method, and (iii) the final result involves quantities which are not exponentially small in the noise strength. This final result, which gives the probability of a particular state being selected in terms of the parameters of the dynamics, is remarkably simple and agrees well with the results of numerical simulations. The method should be applicable to similar problems in a number of other areas such as state selection in lasers, activationless chemical reactions and population dynamics in fluctuating environments.Comment: 28 pages, 6 figures. Accepted for publication in Phys. Rev.

A unified approach for the solution of the Fokker-Planck equation

This paper explores the use of a discrete singular convolution algorithm as a unified approach for numerical integration of the Fokker-Planck equation. The unified features of the discrete singular convolution algorithm are discussed. It is demonstrated that different implementations of the present algorithm, such as global, local, Galerkin, collocation, and finite difference, can be deduced from a single starting point. Three benchmark stochastic systems, the repulsive Wong process, the Black-Scholes equation and a genuine nonlinear model, are employed to illustrate the robustness and to test accuracy of the present approach for the solution of the Fokker-Planck equation via a time-dependent method. An additional example, the incompressible Euler equation, is used to further validate the present approach for more difficult problems. Numerical results indicate that the present unified approach is robust and accurate for solving the Fokker-Planck equation.Comment: 19 page

Dichotomous Markov noise: Exact results for out-of-equilibrium systems. A review

Nonequilibrium systems driven by additive or multiplicative dichotomous Markov noise appear in a wide variety of physical and mathematical models. We review here some prototypical examples, with an emphasis on {\em analytically-solvable} situations. In particular, it has escaped attention till recently that the standard results for the long-time properties of such systems cannot be applied when unstable fixed points are crossed in the asymptotic regime. We show how calculations have to be modified to deal with these cases and present a few relevant applications -- the hypersensitive transport, the rocking ratchet, and the stochastic Stokes' drift. These results reinforce the impression that dichotomous noise can be put on a par with Gaussian white noise as far as obtaining analytical results is concerned. They convincingly illustrate the interplay between noise and nonlinearity in generating nontrivial behaviors of nonequilibrium systems and point to various practical applications.Comment: Review article, 85 pages, 24 figures. Some references added. Published in the International Journal of Modern Physics B 20, pp. 2825-2888 (2006). Comments and suggestions from the readers are highly welcom

Taxation in the digital economy - recent policy developments and the question of value creation

The paper reviews the evidence on the challenges of digitalization for direct (corporate profit) and indirect (consumption) taxation. Based on both anecdotal and empirical evidence, we evaluate ongoing developments at the OECD and European Union level and argue that there is no justification for introducing a new tax order for digital businesses. In particular, the significant digital presence and the digital services tax as put forward by the European Commission will most likely distort corporate decisions and spur tax competition. To contribute to the development of tax rules in line with value creation as the gold standard for profit taxation the paper discusses data as a "new" value-driving asset in the digital economy. It draws on insights from interdisciplinary research to highlight that the value of data emerges through proprietary activities conducted within businesses. We ultimately discuss how existing transfer pricing solutions can be adapted to business models employing data mining

Probability Theory in Statistical Physics, Percolation, and Other Random Topics: The Work of C. Newman

In the introduction to this volume, we discuss some of the highlights of the research career of Chuck Newman. This introduction is divided into two main sections, the first covering Chuck's work in statistical mechanics and the second his work in percolation theory, continuum scaling limits, and related topics.Comment: 38 pages (including many references), introduction to Festschrift in honor of C.M. Newma

Study of a class of models for self-organization: equilibrium analysis

A new class of nonlinear stochastic models is introduced with a view to explore self-organization. The model consists of an assembly of anharmonic oscillators, interacting via a mean field of system size range, in presence of white, Gaussian noise. Its properties are explored in the overdamped regime (Smoluchowski limit). The single oscillator potential is such that for small oscillator displacements it leads to a highly nonlinear force but becomes asymptotically harmonic. The shape of the potential can be a single-or double-well and is controlled by a set of parameters. Through equilibrium statistical mechanical analysis, we study the collective behavior and the nature of phase transition. Much of the analysis is analytic and exact. The treatment is not restricted to the thermodynamic limit so that we are also able to discuss finite size effects in the model. © 1983 Plenum Publishing Corporation