Universal first-passage properties of discrete-time random walks and Lévy flights on a line: Statistics of the global maximum and records

In these lecture notes I will discuss the universal first-passage properties of a simple correlated discrete-time sequence {x₀=0,x₁,x₂,…,x_n} up to n steps where x_i represents the position at step i of a random walker hopping on a continuous line by drawing independently, at each time step, a random jump length from an arbitrary symmetric and continuous distribution (it includes, e.g., the Lévy flights). I will focus on the statistics of two extreme observables associated with the sequence: (i) its global maximum and the time step at which the maximum occurs and (ii) the number of records in the sequence and their ages. I will demonstrate how the universal statistics of these observables emerge as a consequence of Pollaczek-Spitzer formula and the associated Sparre Andersen theorem.     

An invariance principle for reversible Markov processes. Applications to random motions in random environments

We present an invariance principle for antisymmetric functions of a reversible Markov process which immediately implies convergence to Brownian motion for a wide class of random motions in random environments. We apply it to establish convergence to Brownian motion (i) for a walker moving in the infinite cluster of the two-dimensional bond percolation model, (ii) for ad-dimensional walker moving in a symmetric random environment under very mild assumptions on the distribution of the environment, (iii) for a tagged particle in ad-dimensional symmetric lattice gas which allows interchanges, (iv) for a tagged particle in ad-dimensional system of interacting Brownian particles. Our formulation also leads naturally to bounds on the diffusion constant.     

Distribution functions for random walk processes on networks: An analytic method

A novel analytic method for deriving and analyzing probability distribution functions of variables arising in random walk problems is presented. Applications of the method to quasi-one-dimensional systems show that the generating functions of interest possess simple poles, and no branch cuts outside the unit complex disk. This fact makes it possible to derive closed formulas for the full probability distribution functions and to analyze their properties. We find that transverse structures attached to a one-dimensional backbone can be responsible for the appearance of power laws in observables such as the distribution of first arrival times or the total current moving through a (model) photoexcited dirty semiconductor (our results compare well with experiment). We conclude that in some cases a geometrical effect, e.g., that of a transverse structure, may be indistinguishable from a dynamical effect (long waiting time); we also find universal shapes of distribution functions (humped structures) which are not characterized by power laws. The role of bias in determining properties of quasi-one-dimensional structures is examined. A master equation for generating functions is derived and applied to the computation of currents. Our method is also applied to a fractal structure, yielding nontrivial power laws. In all finite networks considered, all probability distributions decay exponentially for asymptotically long times.For a relatively recent review with some historical background see Ref. 2.