Entry times in automata with simple defect dynamics

TL;DR

This paper analyzes the asymptotic distribution of entry times in a simple cellular automaton with two particles, using probabilistic approximations to derive explicit results for various initial conditions.

Contribution

It introduces a method to approximate the automaton's behavior with Brownian motion, extending previous work to a broader class of initial measures and similar automata.

Findings

Explicit asymptotic distribution formulas for entry times

Approximation of automaton dynamics with Brownian motion

Applicability to a wide class of initial measures and automata

Abstract

In this paper, we consider a simple cellular automaton with two particles of different speeds that annihilate on contact. Following a previous work by K\r urka et al., we study the asymptotic distribution, starting from a random configuration, of the waiting time before a particle crosses the central column after time n. Drawing a parallel between the behaviour of this automata on a random initial configuration and a certain random walk, we approximate this walk using a Brownian motion, and we obtain explicit results for a wide class of initial measures and other automata with similar dynamics.