Random and deterministic walks on lattices

TL;DR

This paper reviews classical and non-classical walks on lattices, highlighting how different microscopic dynamics lead to diverse diffusive and organized behaviors relevant in various physical systems.

Contribution

It provides a comprehensive review of both stochastic and deterministic lattice walks, emphasizing their physical implications and the variety of behaviors they can produce.

Findings

Classical random walks model diffusive processes effectively.

Deterministic walks can produce directed propagation and organization.

Lattice dynamics approach helps understand diverse physical phenomena.

Abstract

Random walks of particles on a lattice are a classical paradigm for the microscopic mechanism underlying diffusive processes. In deterministic walks, the role of space and time can be reversed, and the microscopic dynamics can produce quite different types of behavior such as directed propagation and organization, which appears to be generic behaviors encountered in an important class of systems. The various aspects of classical and not so classical walks on latices are reviewed with emphasis on the physical phenomena that can be treated through a lattice dynamics approach.