Walks In Random Media Research Articles

Symmetric branching random walks in random media: comparing theoretical and numerical results

We consider a continuous-time branching random walk on a multidimensional lattice in a random branching medium. It is theoretically known that, in such branching random walks, large rare fluctuations of the medium may lead to anomalous properties of a particle field, e.g., such as intermittency. However, the time intervals on which this intermittency phenomenon can be observed are very difficult to estimate in practice. In this paper, branching media containing only a finite and non-finite number of branching sources are considered. The evolution of the mean number of particles with a random point perturbation and one initial ancestor particle at a lattice point is described by an appropriate Cauchy problem for the evolutionary operator. We review some previous results about the long-time behavior of the medium-averaged moments for the particle population at every lattice point as well as the total one over the lattice and present an algorithm for the simulation of branching random walks under various assumptions about the medium, including the medium randomness. The effects arising in random non-homogeneous and homogeneous media are then compared and illustrated by simulations based on the potential with Weibull-type upper tail. A wide range of models under different assumptions on a branching medium, a configuration of branching sources, and a lattice dimension were considered during the comparison. The simulation results indicate that intermittency can be observed in random media even over finite time intervals.

Heat Kernels, Stochastic Processes and Functional Inequalities

The general topic of the 2013 workshop Heat kernels, stochastic processes and functional inequalities was the study of linear and non-linear diffusions in geometric environments: finite and infinite-dimensional manifolds, metric spaces, fractals and graphs, including random environments. The workshop brought together leading researchers from analysis, probability and geometry and provided a unique opportunity for interaction of established and young scientists from these areas. Unifying themes were heat kernel analysis, mass transport problems and related functional inequalities such as Poincar´e, Sobolev, logarithmic Sobolev, Bakry-Emery, Otto-Villani and Talagrand inequalities. These concepts were at the heart of Perelman’s proof of Poincar´e’s conjecture, as well as of the development of the Otto calculus, and the synthetic Ricci bounds of Lott-Sturm-Villani. The workshop provided participants with an opportunity to discuss how these techniques can be used to approach problems in optimal transport for non-local operators, subelliptic operators in finite and infinite dimensions, analysis on singular spaces, as well as random walks in random media.

Influence of memory in deterministic walks in random media: Analytical calculation within a mean-field approximation

Consider a random medium consisting of N points randomly distributed so that there is no correlation among the distances separating them. This is the random link model, which is the high dimensionality limit (mean-field approximation) for the Euclidean random point structure. In the random link model, at discrete time steps, a walker moves to the nearest point, which has not been visited in the last mu steps (memory), producing a deterministic partially self-avoiding walk (the tourist walk). We have analytically obtained the distribution of the number n of points explored by the walker with memory mu=2 , as well as the transient and period joint distribution. This result enables us to explain the abrupt change in the exploratory behavior between the cases mu=1 (memoryless walker, driven by extreme value statistics) and mu=2 (walker with memory, driven by combinatorial statistics). In the mu=1 case, the mean newly visited points in the thermodynamic limit (N1) is just n=e=2.72... while in the mu=2 case, the mean number n of visited points grows proportionally to N;{12} . Also, this result allows us to establish an equivalence between the random link model with mu=2 and random map (uncorrelated back and forth distances) with mu=0 and the abrupt change between the probabilities for null transient time and subsequent ones.

Deterministic and random partially self-avoiding walks in random media

Consider a set of N cities randomly distributed in the bulk of a hypercube with d dimensions. A walker, with memory μ , begins his route from a given city of this map and moves, at each discrete time step, to the nearest point, which has not been visited in the preceding μ steps. After reviewing the more interesting general results, we consider one-dimensional disordered media and show that the walker needs not to have full memory of its trajectory to explore the whole system, it suffices to have memory of order ln N / ln 2 .

Deterministic Walks in Random Media

Deterministic walks over a random set of N points in one and two dimensions ( d = 1,2) are considered. Points ("cities") are randomly scattered in R(d) following a uniform distribution. A walker ("tourist"), at each time step, goes to the nearest neighbor city that has not been visited in the past tau steps. Each initial city leads to a different trajectory composed of a transient part and a final p-cycle attractor. Transient times (for d = 1,2) follow an exponential law with a tau-dependent decay time but the density of p cycles can be approximately described by D(p)proportional to p(-alpha(tau)). For tau>>1 and tau/N<<1, the exponent is independent of tau. Some analytical results are given for the d = 1 case.

Directed self-avoiding walks in random media.

Two types of directed self-avoiding walks (SAW's), namely, three-choice directed SAW and outwardly directed SAW, have been studied on infinite percolation clusters on the square lattice in two dimensions. The walks on the percolation clusters are generated via a Monte Carlo technique. The longitudinal extension R(N) and the transverse fluctuation W(N) have been measured as a function of the number of steps N. Slight swelling is observed in the longitudinal direction on the random lattices. A crossover from shrinking to swelling of the transverse fluctuations is found at a certain length N(c) of the walks. The exponents related to the transverse fluctuations are seen to be unchanged in the random media even as the percolation threshold is reached. The scaling function form of the extensions are verified.

Random Walks in Random Media on a Cayley Tree

We present sufficient conditions for the transience of random walks with bounded jumps in random media on a Cayley tree.

Field theory of self-avoiding walks in random media

Based on the analogy with the quantum mechanics of a particle propagating in a complex potential, we develop a field-theoretical description of the statistical properties of a self-avoiding polymer chain in a random environment. We show that the account of the non-Hermiticity of the quantum Hamiltonian results in a qualitatively different structure of the effective action, compared to previous studies. Applying renormalization group analysis, we find a transition between the weak-disorder regime, where the quenched randomness is irrelevant, and the strong-disorder regime, where the polymer chain collapses. However, the fact that the renormalized interaction constants and the chiral symmetry breaking regularization parameter flow towards strong coupling raises questions about the applicability of the perturbative analysis.

Reply

We are in general agreement with the views expressed by Rusakov and Kullmann in their letter and would add only the following: Tortuosity can be simply defined as λ=(D/D*)0.5 where D and D* are the diffusion coefficients in water and in tissue, respectively. However, relating this definition to local geometrical properties raises deep theoretical issues about measures of ensembles of random walks in random media and involves both local structure and connectivity of the tissue. In addition, viscosity, particle capture (uptake) and other physical effects might come into play, rendering λ a composite parameter. Our review[ 1 Nicholson C. Syková E. Trends Neurosci. 1998; 21: 207-215 Abstract Full Text Full Text PDF PubMed Scopus (739) Google Scholar ]shows, and the comments of Rusakov and Kullmann further support the argument, that we now have enough experimental data to enable a more sophisticated analysis of tortuosity. Indeed, recent experiments in our laboratories in New York and in Prague, employing osmotic stress, substantiate the existence of at least two components of tortuosity.

A hierarchical model for random walks in random media

We show that the random walk generated by a hierarchical Laplacian in ℤd has standard diffusive behavior. Moreover, we show that this behavior is stable under a class of random perturbations that resemble an off-diagonal disordered lattice Laplacian. The density of states and its asymptotic behavior around zero energy are computed: singularities appear in one and two dimensions.

Random walks in random media with random signs

Monte Carlo calculations indicate that the transverse displacement and free energy fluctuations of 1 + 1 dimensional directed random walks in media with sign randomness are characterized by the same exponents as directed walks in ordinary random media, with positive statistical weights.