Exact Spatio-Temporal Dynamics of Lattice Random Walks in Hexagonal and Honeycomb Domains #

Daniel Marris

13:30 Tuesday in 3Q16.

Part of the Stochastic processes and random walks session.

Abstract #

Lattice random walks in hexagonal domains are employed to model the dynamics of stochastic systems in a wide range of applications ranging from the movement of mobile users in communication networks to the formation of territories in scent-marking animals. As lattice walkers in finite hexagonal geometries are subject to complicated ‘zig-zag’ boundary conditions, analytic representation of their dynamics have, so far, not been accessible. By generalising the method of images to hexagonal geometries, we obtain closed-form expressions for the occupation probability, the so-called propagator, for lattice random walks both on six-neighbour (hexagonal) and three-neighbour (honeycomb) lattices with periodic boundary conditions. Utilising the so-called defect-technique, we then construct the exact propagators for the absorbing and reflecting boundary conditions and derive transport-related statistical quantities such as first passage probabilities to one or multiple targets and their means.