Random Walk on the Range of Random Walk

We study the random walk X on the range of a simple random walk on ℤ ^d in dimensions d≥4. When d≥5 we establish quenched and annealed scaling limits for the process X, which show that the intersections of the original simple random walk path are essentially unimportant. For d=4 our results are less precise, but we are able to show that any scaling limit for X will require logarithmic corrections to the polynomial scaling factors seen in higher dimensions. Furthermore, we demonstrate that when d=4 similar logarithmic corrections are necessary in describing the asymptotic behavior of the return probability of X to the origin.     

Fractal behavior in trapping and reaction: A random walk study

We investigate the trapping of a random walker in fractal structures (Sierpinski gaskets) with randomly distributed traps. The survival probability is determined from the number of distinct sites visited in the trap-free fractals. We show that the short-time behavior and the long-time tails of the survival probability are governed by the spectral dimensiond. We interpolate between these two limits by introducing a scaling law. An extension of the theory, which includes a continuous-time random walk on fractals, is discussed as well as the case of direct trapping. The latter case is shown to be governed by the fractal dimensiond.     

Temperature controlled Lévy flights of minority carriers in photoexcited bulk n-InP

We study by photoluminescence the spatial distribution of minority carriers (holes) arising from their anomalous photon-assisted diffusion upon photo-excitation at an edge of n-InP slab for temperatures ranging from 300 K to 78 K. Giant enhancement in the spread of holes — over distances exceeding 1 cm from the excitation edge — is seen at lower temperatures. We show that the experiment provides a realization of the “Lévy flight” random walk of holes, in which the Lévy distribution index γ   is controlled by the temperature. The variation γ(T)$\gamma (T)$ is close to that predicted earlier, γ=1−Δ/kT$\gamma =1-\Delta /kT$, where Δ(T)$\Delta (T)$ is the Urbach tailing parameter of the absorption spectra. This theoretical prediction is based on the assumption of a quasi-equilibrium intrinsic emission spectrum in the form due to van Roosbroeck and Shockley.     

Statistics of the Occupation Time of Renewal Processes

We present a systematic study of the statistics of the occupation time and related random variables for stochastic processes with independent intervals of time. According to the nature of the distribution of time intervals, the probability density functions of these random variables have very different scalings in time. We analyze successively the cases where this distribution is narrow, where it is broad with index <1, and finally where it is broad with index 1<<2. The methods introduced in this work provide a basis for the investigation of the statistics of the occupation time of more complex stochastic processes (see joint paper by G. De Smedt, C. Godrèche, and J. M. Luck⁽²⁶⁾).     

Continuous-Time Classical and Quantum Random Walk on Direct Product of Cayley Graphs

In this paper we define direct product of graphs and give a recipe for obtaining probability of observing particle on vertices in the continuous-time classical and quantum random walk. In the recipe, the probability of observing particle on direct product of graph is obtained by multiplication of probability on the corresponding to sub-graphs, where this method is useful to determining probability of walk on compficated graphs. Using this method, we calculate the probability of Continuous-time classical and quantum random walks on many of finite direct product Cayley graphs （complete cycle, complete Kn, charter and n-cube）. Also, we inquire that the classical state the stationary uniform distribution is reached as t→∞ but for quantum state is not always satisfied.     

Heavy-tailed distributions in the intermittent motion behaviour of the intertidal gastropod Littorina littorea

The two-dimensional motion behaviour of the common intertidal gastropod Littorina littorea is investigated as a function of the immersion time from three sampling sites on an exposed rocky shore. A total of 90 individuals have been individually marked and tracked over 14 consecutive daylight low tide. Successive displacements show very intermittent behaviour, with a few localised large displacements over a wide range of small displacements. We show that successive displacements are described by flight length l_d heavy-tailed distributions with . The very low values of the exponent μ (μ≈2.22, 2.43 and 2.67) indicate that L. littorea flights fall into the category of super-diffusive processes. These exponents were significantly higher than the special value μ≈2 analytically and theoretically predicted to be the most advantageous in optimising long-term encounter statistics, especially for low-prey-density scenario. As natural selection should favour flexible behaviour, leading to different optimum searching statistics, under different conditions, our results support the idea that the differences in food concentration and distribution encountered at the different sites by L. littorea led to different heavy-tailed distributions observed for the most extreme displacements.     

On the first passage time and leapover properties of Lévy motions

We investigate two coupled properties of Lévy stable random motions: the first passage times (FPTs) and the first passage leapovers (FPLs). While, in general, the FPT problem has been studied quite extensively, the FPL problem has hardly attracted any attention. Considering a particle that starts at the origin and performs random jumps with independent increments chosen from a Lévy stable probability law λ_α,β(x), the FPT measures how long it takes the particle to arrive at or cross a target. The FPL addresses a different question: given that the first passage jump crosses the target, then how far does it get beyond the target? These two properties are investigated for three subclasses of Lévy stable motions: (i) symmetric Lévy motions characterized by Lévy index α(0<α<2) and skewness parameter β=0, (ii) one-sided Lévy motions with 0<α<1, β=1, and (iii) two-sided skewed Lévy motions, the extreme case, 1<α<2, β=−1.     

The emergence of scaling laws search dynamics in a particle swarm optimization

This paper investigates the search dynamics of a fundamental particle swarm optimization (PSO) algorithm via gathering and analyzing the data of the search area during the optimization process. The PSO algorithm exhibits a distinct performance when optimizing different functions, which induces the emergence of different search dynamics during the optimization process. The simulation results show that the performance is tightly related to the search dynamics which results from the interaction between the PSO algorithm and the landscape of the solved problems. The Lévy type scaling laws search dynamics emerges from the process in which the PSO algorithm shows good performance, while the Brownian dynamics appears after the algorithm has stagnated due to the premature convergence. The Lévy dynamics characterized by a large number of intensive local searches punctuated by long-range transfers is an indicator of good performance, which allows the algorithm to achieve an efficient balance between exploration and exploitation so as to improve the search efficiency.     

Return Probability of the Fibonacci Quantum Walk

In this paper the return probability of the one-dimensional discrete-time quantum walk is studied. We derive probabilistic formulas for the return probability related to the quantum walk governed by the Fibonacci coin.     

Return Probability of the Fibonacci Quantum Walk

In this paper the return probability of the one-dimensional discrete-time quantum walk is studied. We derive probabilistic formulas for the return probability related to the quantum walk governed by the Fibonacci coin.     

General Asymptotics of the Density of a Restricted Coalescing Random Walk System

These results explore the asymptotic behavior of the density of a system of coalescing random walks where particles begin from only a subspace of the integer lattice and are allowed to walk anywhere on the lattice. They generalize results by Bramson and Griffeath from 1980.⁽¹⁾ Since the probability that a given site is occupied depends on how far that site is from the originating subspace, the density of the system at a given time must be re-defined. However, the general idea is still that if the density is larger than we expect at a given time, more coalescing events will occur, and the density will correct itself over time.     

On the origin of bursts and heavy tails in animal dynamics

Over recent years there has been an accumulation of evidence that many animal behaviours are characterised by common scale-invariant patterns of switching between two contrasting activities over a period of time. This is evidenced in mammalian wake-sleep patterns, in the intermittent stop-start locomotion of Drosophila fruit flies, and in the Lévy walk movement patterns of a diverse range of animals in which straight-line movements are punctuated by occasional turns. Here it is shown that these dynamics can be modelled by a stochastic variant of Barabási’s model [A.-L. Barabási, The origin of bursts and heavy tails in human dynamics, Nature 435 (2005) 207-211] for bursts and heavy tails in human dynamics. The new model captures a tension between two competing and conflicting activities. The durations of one type of activity are distributed according to an inverse-square power-law, mirroring the ubiquity of inverse-square power-law scaling seen in empirical data. The durations of the second type of activity follow exponential distributions with characteristic timescales that depend on species and metabolic rates. This again is a common feature of animal behaviour. Bursty human dynamics, on the other hand, are characterised by power-law distributions with scaling exponents close to −1 and −3/2.     

Balancing the competing demands of harvesting and safety from predation: Lévy walk searches outperform composite Brownian walk searches but only when foraging under the risk of predation

Some foragers have movement patterns that can be approximated by Lévy walks whilst others may be better represented as composite Brownian walks. Many attempts have been made to interpret these movement patterns in terms of optimal searching strategies for the location of randomly and sparsely distributed targets. Here it is shown that the relative merits of Lévy walk and composite Brownian walk searches are sensitively dependent upon target encounter dynamics which set the initial conditions for an extensive search. It is suggested these initial conditions are determined, at least in part, by the competing demands of harvesting and safety from predation. In accordance with observations, it is shown that Lévy walks are expected in tritrophic systems and where intraguild predation operates. Composite Brownian walks, on the other hand, are found to be advantageous when the risk of predation is low. Despite having fundamentally different properties, Lévy walks and composite Brownian walks can therefore compete a priori as possible models of animal movements. Throughout, attention is focused on searching for randomly and sparsely distributed resources that are not depleted or rejected once located but instead remain targets for future searches. We re-evaluate and overturn the widely held belief that in numerical simulations this ‘non-destructive’ searching scenario can faithfully and consistently represent destructive searching for patchily distributed resources, i.e. for resources that tend to occur in clusters rather than in isolation.     

Time evolution of the probability distribution in stochastic and chaotic systems with enhanced diffusion

We perform a detailed study of the time evolution of the probability distribution for two processes displaying enhanced diffusion: a stochastic process named the Lévy walk and a deterministic chaotic process, the amplified climbing-sine map. The time evolution of the probability distribution differs in the two cases and carries information which is peculiar to the investigated process.     

Return Probability of the Open Quantum Random Walk with Time-Dependence

We study the open quantum random walk (OQRW) with time-dependence on the one-dimensional lattice space and obtain the associated limit distribution. As an application we study the return probability of the OQRW. We also ask, "What is the average time for the return probability of the OQRW?"     

Energy and number of collision fluctuations in inelastic gases

The two-dimensional Inelastic Maxwell Model (IMM) is studied by numerical simulations. It is shown how the inelasticity of collisions together with the fluctuations of the number of collisions undergone by a particle lead to energy fluctuations. These fluctuations are associated to a shrinking of the available phase space. We find the asymptotic scaling of these energy fluctuations and show how they affect the tail of the velocity distribution during long time intervals. We stress that these fluctuations relax like power laws on much slower time scales than the usual exponential relaxations taking place in kinetic theory.     

The influence of the environment on Lévy random search efficiency: Fractality and memory effects

An open problem in the field of random searches relates to optimizing the search efficiency in fractal environments. Here we address this issue through a systematic study of Lévy searches in landscapes encompassing several degrees of target aggregation and fractality. For scarce resources, non-destructive searches with unrestricted revisits to targets are shown to present universal optimal behavior irrespective of the general scaling properties of the spatial distribution of targets. In contrast, no such universal behavior occurs in the destructive case with forbidden revisits, in which the optimal strategy strongly depends on the degree of target aggregation. We also investigate how the presence of memory and learning skills of the searcher affect the search efficiency. By considering a limiting model in which the searcher learns through recent experience to recognize food-rich areas, we find that a statistical memory of previous encounters does not necessarily increase the rate of target findings in random searches. Instead, there is an optimal extent of memory, dependent on specific details of the search space and stochastic dynamics, which maximizes the search efficiency. This finding suggests a more general result, namely that in some instances there are actual advantages to ignoring certain pieces of partial information while searching for objects.     

Measuring the self-similarity exponent in Lévy stable processes of financial time series

Geometric method-based procedures, which will be called GM algorithms herein, were introduced in [M.A. Sánchez Granero, J.E. Trinidad Segovia, J. García Pérez, Some comments on Hurst exponent and the long memory processes on capital markets, Phys. A 387 (2008) 5543-5551], to efficiently calculate the self-similarity exponent of a time series. In that paper, the authors showed empirically that these algorithms, based on a geometrical approach, are more accurate than the classical algorithms, especially with short length time series. The authors checked that GM algorithms are good when working with (fractional) Brownian motions. Moreover, in [J.E. Trinidad Segovia, M. Fernández-Martínez, M.A. Sánchez-Granero, A note on geometric method-based procedures to calculate the Hurst exponent, Phys. A 391 (2012) 2209-2214], a mathematical background for the validity of such procedures to estimate the self-similarity index of any random process with stationary and self-affine increments was provided. In particular, they proved theoretically that GM algorithms are also valid to explore long-memory in (fractional) Lévy stable motions.