Site-bond representation and self-duality for totalistic probabilistic cellular automata

TL;DR

This paper investigates one-dimensional totalistic probabilistic cellular automata with long-range interactions, establishing conditions for site-bond representation, self-duality, and convergence, extending the Domany-Kinzel model.

Contribution

It introduces new conditions for site-bond representation and self-duality in TPCA with long-range interactions, extending existing models.

Findings

Conditions for site-bond representation are established.

Self-duality conditions are derived and expressed mathematically.

A convergence theorem for the TPCA is proved.

Abstract

We study the one-dimensional two-state totalistic probabilistic cellular automata (TPCA) having an absorbing state with long-range interactions, which can be considered as a natural extension of the Domany-Kinzel model. We establish the conditions for existence of a site-bond representation and self-dual property. Moreover we present an expression of a set-to-set connectedness between two sets, a matrix expression for a condition of the self-duality, and a convergence theorem for the TPCA.