Propagators of random walks on comb lattices of arbitrary dimension

TL;DR

This paper derives exact formulas for the propagators of random walks on comb lattices of any dimension, providing insights into diffusion processes and mean-square displacement calculations.

Contribution

It introduces explicit Laplace transform formulas for propagators on arbitrary-dimensional comb lattices using first-passage properties and a master equation approach.

Findings

Exact Laplace transform formulas for propagators

Explicit mean-square displacement calculations

Extension to complex comb-like structures

Abstract

We study diffusion on comb lattices of arbitrary dimension. Relying on the loopless structure of these lattices and using first-passage properties, we obtain exact and explicit formulae for the Laplace transforms of the propagators associated to nearest-neighbour random walks in both cases where either the first or the last point of the random walk is on the backbone of the lattice, and where the two extremities are arbitrarily chosen. As an application, we compute the mean-square displacement of a random walker on a comb of arbitrary dimension. We also propose an alternative and consistent approach of the problem using a master equation description, and obtain simple and generic expressions of the propagators. This method is more general and is extended to study the propagators of random walks on more complex comb-like structures. In particular, we study the case of a two-dimensional comb lattice with teeth of finite length.