Levy Flight

We generalize the continuous time random walk (CTRW) to include the effect of space dependent jump probabilities. When the mean waiting time diverges we derive a fractional Fokker-Planck equation (FFPE). This equation describes anomalous diffusion in an external force field and close to thermal equilibrium. We discuss the domain of validity of the fractional kinetic equation. For the force free case we compare between the CTRW solution and that of the FFPE.

Shenvi, Kempe and Whaley's quantum random-walk search (SKW) algorithm [Phys. Rev. A 67, 052307 (2003)] is known to require $O(\sqrt N)$ number of oracle queries to find the marked element, where $N$ is the size of the search space. The overall time complexity of the SKW algorithm differs from the best achievable on a quantum computer only by a constant factor. We present improvements to the SKW algorithm which yield significant increase in success probability, and an improvement on query complexity such that the theoretical limit of a search algorithm succeeding with probability close to one is reached. We point out which improvement can be applied if there is more than one marked element to find.

We analyze the sequence of time intervals between consecutive stock trades of thirty companies representing eight sectors of the U. S. economy over a period of four years. For all companies we find that: (i) the probability density function of intertrade times may be fit by a Weibull distribution; (ii) when appropriately rescaled the probability densities of all companies collapse onto a single curve implying a universal functional form; (iii) the intertrade times exhibit power-law correlated behavior within a trading day and a consistently greater degree of correlation over larger time scales, in agreement with the correlation behavior of the absolute price returns for the corresponding company, and (iv) the magnitude series of intertrade time increments is characterized by long-range power-law correlations suggesting the presence of nonlinear features in the trading dynamics, while the sign series is anti-correlated at small scales. Our results suggest that independent of industry sector, market capitalization and average level of trading activity, the series of intertrade times exhibit possibly universal scaling patterns, which may relate to a common mechanism underlying the trading dynamics of diverse companies. Further, our observation of long-range power-law correlations and a parallel with the crossover in the scaling of absolute price returns for each individual stock, support the hypothesis that the dynamics of transaction times may play a role in the process of price formation.

We study structure, eigenvalue spectra and diffusion dynamics in a wide class of networks with subgraphs (modules) at mesoscopic scale. The networks are grown within the model with three parameters controlling the number of modules, their internal structure as scale-free and correlated subgraphs, and the topology of connecting network. Within the exhaustive spectral analysis for both the adjacency matrix and the normalized Laplacian matrix we identify the spectral properties which characterize the mesoscopic structure of sparse cyclic graphs and trees. The minimally connected nodes, clustering, and the average connectivity affect the central part of the spectrum. The number of distinct modules leads to an extra peak at the lower part of the Laplacian spectrum in cyclic graphs. Such a peak does not occur in the case of topologically distinct tree-subgraphs connected on a tree. Whereas the associated eigenvectors remain localized on the subgraphs both in trees and cyclic graphs. We also find a characteristic pattern of periodic localization along the chains on the tree for the eigenvector components associated with the largest eigenvalue equal 2 of the Laplacian. We corroborate the results with simulations of the random walk on several types of networks. Our results for the distribution of return-time of the walk to the origin (autocorrelator) agree well with recent analytical solution for trees, and it appear to be independent on their mesoscopic and global structure. For the cyclic graphs we find new results with twice larger stretching exponent of the tail of the distribution, which is virtually independent on the size of cycles. The modularity and clustering contribute to a power-law decay at short return times.

Visual attention guides our gaze to relevant parts of the viewed scene, yet the moment-to-moment relocation of gaze can be different among observers even though the same locations are taken into account. Surprisingly, the variability of eye movements has been so far overlooked by the great majority of computational models of visual attention.
In this paper we present the Ecological Sampling model, a stochastic model of eye guidance explaining such variability. The gaze shift mechanism is conceived as an active random sampling that the ”foraging eye” carries out upon the visual landscape, under the constraints set by the observable features and the global complexity of the landscape. By drawing on results reported in the foraging literature, the actual gaze relocation is eventually driven by a stochastic differential equation whose noise source is sampled from a mixture of α-stable distributions.
This way, the sampling strategy proposed here allows to mimic a fundamental property of the eye guidance mechanism: where we choose to look next at any given moment in time is not completely deterministic, but neither is it completely random
To show that the model yields gaze shift motor behaviors that exhibit statistics similar to those displayed by human observers, we compare simulation outputs with those obtained from eye- tracked subjects while viewing complex dynamic scenes

Based on the Langevin description of the Continuous Time Random Walk (CTRW), we consider a generalization of CTRW in which the waiting times between the subsequent jumps are correlated. We discuss the cases of exponential and slowly decaying persistent power-law correlations between the waiting times as two generic examples and obtain the corresponding mean squared displacements as functions of time. In the case of exponential-type correlations the (sub)diffusion at short times is slower than in the absence of correlations. At long times the behavior of the mean squared displacement is the same as in uncorrelated CTRW. For power-law correlations we find subdiffusion characterized by the same exponent at all times, which appears to be smaller than the one in uncorrelated CTRW. Interestingly, in the limiting case of an extremely long power-law correlations, the (sub)diffusion exponent does not tend to zero, but is bounded from below by the subdiffusion exponent corresponding to a short time behavior in the case of exponential correlations.

We present an exact and Monte Carlo renormalization group (MCRG) study of semiflexible polymer chains on an infinite family of the plane-filling (PF) fractals. The fractals are compact, that is, their fractal dimension df is equal to 2 for all members of the fractal family enumerated by the odd integer b(3≤b<∞) . For various values of stiffness parameter s of the chain, on the PF fractals (for 3≤b≤9 ), we calculate exactly the critical exponents ν (associated with the mean squared end-to-end distances of polymer chain) and γ (associated with the total number of different polymer chains). In addition, we calculate ν and γ through the MCRG approach for b up to 201. Our results show that for each particular b , critical exponents are stiffness dependent functions, in such a way that the stiffer polymer chains (with smaller values of s ) display enlarged values of ν , and diminished values of γ . On the other hand, for any specific s , the critical exponent ν monotonically decreases, whereas the critical exponent γ monotonically increases, with the scaling parameter b . We reflect on a possible relevance of the criticality of semiflexible polymer chains on the PF family of fractals to the same problem on the regular Euclidean lattices.

We introduce quantum walks with a time-dependent coin, and show how they include, as a particular case, the generalized quantum walk recently studied by Wojcik et al. {[}Phys. Rev. Lett. \textbf{93}, 180601(2004){]} which exhibits interesting dynamical localization and quasiperiodic dynamics. Our proposal allows for a much easier implementation of this particular rich dynamics than the original one. Moreover, it allows for an additional control on the walk, which can be used to compensate for phases appearing due to external interactions. To illustrate its feasibility, we discuss an example using an optical cavity. We also derive an approximated solution in the continuous limit (long--wavelength approximation) which provides physical insight about the process.

Continuous-time random walk (CTRW) is a model of anomalous sub-diffusion in which particles are immobilized for random times between successive jumps. A power-law distribution of the waiting times, $\psi(\tau) \tau^{-(1+\alpha)}$, leads to sub-diffusion ($<x^2>~t^{\alpha}$) for 0<\alpha<1. In closed systems, the long stagnation periods cause time-averages to divert from the corresponding ensemble averages, which is a manifestation of weak ergodicity breaking. The time-average of a general observable $\bar{U} = \int_0^t U[x(\tau)]d\tau / t$ is a functional of the path and is described by the well known Feynman-Kac equation if the motion is Brownian. Here, we derive forward and backward fractional Feynman-Kac equations for functionals of CTRW in a binding potential. We use our equations to study two specific time-averages: the fraction of time spent by a particle in half box, and the time-average of the particle's position in a harmonic field. In both cases, we obtain the probability density function of the time-averages for $t \rightarrow \infty$ and the first two moments. Our results show that both the occupation fraction and the time-averaged position are random variables even for long-times, except for \alpha=1 when they are identical to their ensemble averages. Using the fractional Feynman-Kac equation, we also study the dynamics leading to weak ergodicity breaking, namely the convergence of the fluctuations to their asymptotic values.