Deterministic walks in random environment

TL;DR

This paper investigates deterministic walks in random environments, demonstrating their reduction to a class of random walks with history-dependent jump probabilities, and establishes foundational probabilistic results for these models.

Contribution

It introduces a reduction of deterministic walks to history-dependent random walks and provides initial probabilistic results for this new class of models.

Findings

Reduction of deterministic walks to history-dependent random walks

Basic probabilistic properties established for the new model

Potential foundation for a general theory of such walks

Abstract

Motivated by the random Lorentz gas, we study deterministic walks in random environment and show that (in simple, yet relevant, cases) they can be reduced to a class of random walks in random environment where the jump probability depends (weakly) on the past. In addition, we prove few basic results (hopefully the germ of a general theory) on the latter, purely probabilistic, model.