Closed-form solutions to the dynamics of confined biased lattice random walks in arbitrary dimensions

Exact spatiotemporal dynamics of lattice random walks in hexagonal and honeycomb domains

A variety of transport processes in natural and man-made systems are intrinsically random. To model their stochasticity, lattice random walks have been employed for a long time, mainly by considering Cartesian lattices. However, in many applications in bounded space the geometry of the domain may have profound effects on the dynamics and ought to be accounted for. We consider here the cases of the six-neighbor (hexagonal) and three-neighbor (honeycomb) lattices, which are utilized in models ranging from adatoms diffusing in metals and excitations diffusing on single-walled carbon nanotubes to animal foraging strategy and the formation of territories in scent-marking organisms. In these and other examples, the main theoretical tool to study the dynamics of lattice random walks in hexagonal geometries has been via simulations. Analytic representations have in most cases been inaccessible, in particular in bounded hexagons, given the complicated "zigzag" boundary conditions that a walker is subject to. Here we generalize the method of images to hexagonal geometries and obtain closed-form expressions for the occupation probability, the so-called propagator, for lattice random walks both on hexagonal and honeycomb lattices with periodic, reflective, and absorbing boundary conditions. In the periodic case, we identify two possible choices of image placement and their corresponding propagators. Using them, we construct the exact propagators for the other boundary conditions, and we derive transport-related statistical quantities such as first-passage probabilities to one or multiple targets and their means, elucidating the effect of the boundary condition on transport properties.

First passage under restart for discrete space and time: Application to one-dimensional confined lattice random walks

First passage under restart has recently emerged as a conceptual framework to study various stochastic processes under restart mechanism. Emanating from the canonical diffusion problem by Evans and Majumdar, restart has been shown to outperform the completion of many first-passage processes which otherwise would take longer time to finish. However, most of the studies so far assumed continuous time underlying first-passage time processes and moreover considered continuous time resetting restricting out restart processes broken up into synchronized time steps. To bridge this gap, in this paper, we study discrete space and time first-passage processes under discrete time resetting in a general setup without specifying their forms. We sketch out the steps to compute the moments and the probability density function which is often intractable in the continuous time restarted process. A criterion that dictates when restart remains beneficial is then derived. We apply our results to a symmetric and a biased random walker in one-dimensional lattice confined within two absorbing boundaries. Numerical simulations are found to be in excellent agreement with the theoretical results. Our method can be useful to understand the effect of restart on the spatiotemporal dynamics of confined lattice random walks in arbitrary dimensions.