Signatures of Lévy flights with annealed disorder

Extreme value statistics of jump processes

We investigate extreme value statistics (EVS) of general discrete time and continuous space symmetric jump processes. We first show that for unbounded jump processes, the semi-infinite propagator G_{0}(x,n), defined as the probability for a particle issued from zero to be at position x after n steps whilst staying positive, is the key ingredient needed to derive a variety of joint distributions of extremes and times at which they are reached. Along with exact expressions, we extract universal asymptotic behaviors of such quantities. For bounded, semi-infinite jump processes killed upon first crossing of zero, we introduce the strip probability μ_{0,[under x]̲}(n), defined as the probability that a particle issued from zero remains positive and reaches its maximum x on its nth^{} step exactly. We show that μ_{0,[under x]̲}(n) is the essential building block to address EVS of semi-infinite jump processes, and obtain exact expressions and universal asymptotic behaviors of various joint distributions.

Topological Constraints on the Dynamics of Vortex Formation in a Two-Dimensional Quantum Fluid

We present experimental and theoretical results on formation of quantum vortices in a laser beam propagating in a nonlinear medium. Topological constrains richer than the mere conservation of vorticity impose an elaborate dynamical behavior to the formation and annihilation of vortex-antivortex pairs. We identify two such mechanisms, both described by the same fold-Hopf bifurcation. One of them is particularly efficient although it is not observed in the context of liquid helium films or stationary systems because it relies on the compressible nature of the fluid of light we consider and on the nonstationarity of its flow.

Leftward, rightward, and complete exit-time distributions of jump processes

First-passage properties of continuous stochastic processes confined in a one-dimensional interval are well described. However, for jump processes (discrete random walks), the characterization of the corresponding observables remains elusive, despite their relevance in various contexts. Here we derive exact asymptotic expressions for the leftward, rightward, and complete exit-time distributions from the interval [0,x] for symmetric jump processes starting from x_{0}=0, in the large x and large time limit. We show that both the leftward probability F_{[under 0]̲,x}(n) to exit through 0 at step n and rightward probability F_{0,[under x]̲}(n) to exit through x at step n exhibit a universal behavior dictated by the large-distance decay of the jump distribution parametrized by the Levy exponent μ. In particular, we exhaustively describe the n≪(x/a_{μ})^{μ} and n≫(x/a_{μ})^{μ} limits and obtain explicit results in both regimes. Our results finally provide exact asymptotics for exit-time distributions of jump processes in regimes where continuous limits do not apply.

Splitting Probabilities of Symmetric Jump Processes

We derive a universal, exact asymptotic form of the splitting probability for symmetric continuous jump processes, which quantifies the probability π_{0,[under x]_}(x_{0}) that the process crosses x before 0 starting from a given position x_{0}∈[0,x] in the regime x_{0}≪x. This analysis provides in particular a fully explicit determination of the transmission probability (x_{0}=0), in striking contrast with the trivial prediction π_{0,[under x]_}(0)=0 obtained by taking the continuous limit of the process, which reveals the importance of the microscopic properties of the dynamics. These results are illustrated with paradigmatic models of jump processes with applications to light scattering in heterogeneous media in realistic 3D slab geometries. In this context, our explicit predictions of the transmission probability, which can be directly measured experimentally, provide a quantitative characterization of the effective random process describing light scattering in the medium.

Sample size effects for Lévy flight of photons in atomic vapors

Lévy flight superdiffusion consists of random walks characterized by very long jumps that dominate the transport. However, the finite size of real samples introduces truncation of long jumps and modifies the transport properties. We measure typical Levy flight parameters for photon diffusion in atomic vapor characterized by a Voigt absorption profile. We observe the change of Lévy parameter as a function of truncation length. We associate this variation with size-dependent contributions from different spectral regions of the emission profile with the Doppler core dominating the transport for thin samples and Lorentz wings for thick samples. Monte Carlo simulations are implemented to support the interpretation of results.

Lévy flights of photons with infinite mean free path

Multiple scattering of light by resonant vapor is characterized by Lévy-type superdiffusion with a single-step size distribution p(x)∝1/x^{1+α}. We investigate Lévy flight of light in a hot rubidium vapor collisional-broadened by 50 torr of He gas. The frequent collisions produce Lorentzian absorptive and emissive profiles with α<1 and a corresponding divergent mean step size. We extract the Lévy parameter α≈0.5 in a multiple-scattering regime from radial profile of the transmission and from violation of the Ohm's law. The measured radial transmission profile and the total diffusive transmission curves are well reproduced by numerical simulations for Lorentzian line shapes.

Generalized time-independent correlation transport equation with static background: influence of anomalous transport on the field autocorrelation function

A generalized time-independent correlation transport equation (GCTE) is proposed for the field autocorrelation function. The GCTE generalizes various models for anomalous transport of photons and takes into account the possible presence of a static background. In a tutorial example, the GCTE is solved for a homogeneous semi-infinite medium in reflectance configuration through Monte Carlo simulations. The chosen anomalous photon transport model also includes the classic and the "generalized" Lambert-Beer's law (depending on the choice of parameters). A numerical algorithm allowing generation of the related anomalous random photon steps is also given. The clear influence of anomalous transport on the field autocorrelation function is shown and discussed for the proposed specific examples by comparing the general results with the classical case (Lambert-Beer's law).

Inferring Lévy walks from curved trajectories: A rescaling method

An important problem in the study of anomalous diffusion and transport concerns the proper analysis of trajectory data. The analysis and inference of Lévy walk patterns from empirical or simulated trajectories of particles in two and three-dimensional spaces (2D and 3D) is much more difficult than in 1D because path curvature is nonexistent in 1D but quite common in higher dimensions. Recently, a new method for detecting Lévy walks, which considers 1D projections of 2D or 3D trajectory data, has been proposed by Humphries et al. The key new idea is to exploit the fact that the 1D projection of a high-dimensional Lévy walk is itself a Lévy walk. Here, we ask whether or not this projection method is powerful enough to cleanly distinguish 2D Lévy walk with added curvature from a simple Markovian correlated random walk. We study the especially challenging case in which both 2D walks have exactly identical probability density functions (pdf) of step sizes as well as of turning angles between successive steps. Our approach extends the original projection method by introducing a rescaling of the projected data. Upon projection and coarse-graining, the renormalized pdf for the travel distances between successive turnings is seen to possess a fat tail when there is an underlying Lévy process. We exploit this effect to infer a Lévy walk process in the original high-dimensional curved trajectory. In contrast, no fat tail appears when a (Markovian) correlated random walk is analyzed in this way. We show that this procedure works extremely well in clearly identifying a Lévy walk even when there is noise from curvature. The present protocol may be useful in realistic contexts involving ongoing debates on the presence (or not) of Lévy walks related to animal movement on land (2D) and in air and oceans (3D).