Random Walk In Presence Research Articles

Residence time in one-dimensional random walks in presence of moving defects

Residence time is relevant in several applicative problems, such as the pedestrian motion in presence of queues and the study of water flows in runoff ponds. In this last context, it is important to control the residence time since it is strictly connected to the pond efficiency in terms of deposition of pollutants. These applications motivated our detailed study of residence time in presence of moving obstacles and defects for a particle performing a one-dimensional random walk. More precisely, for a particle conditioned to exit through the right endpoint, we measure the typical time needed to cross the entire lattice in presence of defects. These are defined as lattice sites with jump probabilities modified adding a bias to the values used at regular sites. We find explicit formulae for the residence time and discuss several models of moving defects. The presence of a stochastic updating rule for the motion of the defect smooths the local residence time profiles found in the case of a static defect.

Quasi-Stationary Regime of a Branching Random Walk in Presence of an Absorbing Wall

A branching random walk in presence of an absorbing wall moving at a constant velocity $v$ undergoes a phase transition as the velocity $v$ of the wall varies. Below the critical velocity $v_c$, the population has a non-zero survival probability and when the population survives its size grows exponentially. We investigate the histories of the population conditioned on having a single survivor at some final time $T$. We study the quasi-stationary regime for $v<v_c$ when $T$ is large. To do so, one can construct a modified stochastic process which is equivalent to the original process conditioned on having a single survivor at final time $T$. We then use this construction to show that the properties of the quasi-stationary regime are universal when $v\to v_c$. We also solve exactly a simple version of the problem, the exponential model, for which the study of the quasi-stationary regime can be reduced to the analysis of a single one-dimensional map.

1‐D random walk in presence of a totally reflecting barrier

AbstractA rigorous analysis of counting paths for 1‐D random walk in the presence of a reflecting barrier is presented. This paper defines and distinguishes between partially and totally reflecting barriers. So far, in the literature only a special case of partially reflecting barrier has been dealt with. An exact combinatorial formula is proven which describes the probability distribution of a diffusing particle at a totally reflecting barrier, allowing computation of any random walk redistribution of a diffusing species near or at the totally reflecting barrier. The analysis shows that for a particle starting its random walk at the barrier, the probability of finding it at the interface is diminishing with the number of diffusion steps N as 1/(N/2 + 1) and that the peak of the probability distribution is moving away from the barrier with the increasing number of steps as . This analysis has implications on the treatment of diffusion of impurities and point defects in thin films and in subsurface layers.

Microscopic derivation of zero-flux boundary condition in one-dimensional random walk in presence of a totally reflecting barrier

In this paper a rigorous probabilistic two-probability-parameter model of a diffusion barrier is investigated that describes comprehensively reflection, absorption, and segregation phenomena at a diffusion barrier. As a special case, a rigorous analysis of counting paths for 1D random walk in the presence of a reflecting barrier is presented. This paper defines and makes distinction between partially and totally reflecting barriers. So far, in the literature only a special case of partially reflecting barrier has been dealt with. A combinatorial formula is derived showing that in the presence of a totally reflecting barrier (at m b=0) the probability of a particle departing from position m=2 j and arriving at position m=2 k on the positive axis after N=2 M steps is given by W 2 j (2 k, N) = [ C(2 M, M− j+ k)− C(2 M, M− j− k− l)]/[ C(2 M, M)+2 ∑ i=0 j C(2 M, M+ i)], where C( n, m) denotes the binomial coefficient. This formula enables easy computation of any random walk redistribution of a diffusing species near or at the totally reflecting barrier. The analysis shows that for a particle starting its random walk at the barrier, the probability of finding it at the interface is diminishing with the number of diffusion steps N=2 M as 1/( M+1) and that the peak of the probability distribution is moving away from the barrier with the increasing number of steps as 4M . Thus, the subsurface region is progressively depleted. The present analysis has bearing on the treatment of diffusion of impurities and point defects in thin films and in subsurface layers.

First passage time distribution in random walks with absorbing boundaries

We calculate the first passage time distribution in simple, unbiased random walks in presence of absorbing boundaries of various shapes. We obtain explicit solutions for the following geometries of the boundaries—a box in one dimension, circular, square and triangular boundaries in two dimensions and cubical box and spherical shell in three dimensions. The distribution in all cases shows scaling and the scaling function can be expressed in terms of the Jacobi Theta functions.

The Galton-Watson Process with Mean One and Finite Variance

The Galton-Watson Process with Mean One and Finite Variance