An example of computation of the density of ones in probabilistic cellular automata by direct recursion

TL;DR

This paper introduces a recursive method to compute the probability of ones in probabilistic cellular automata, demonstrated on a simple diffusion model, enabling exact density calculations from initial random states.

Contribution

The paper presents a novel recursive approach for calculating the density of ones in probabilistic cellular automata, applicable to simple models of information spread.

Findings

Derived explicit formulas for the density of ones.

Validated the recursive method on a diffusion-like PCA.

Showed the method's applicability to simple probabilistic models.

Abstract

We present a method for computing probability of occurence of 1s in a configuration obtained by iteration of a probabilistic cellular automata (PCA), starting from a random initial configuration. If the PCA is sufficiently simple, one can construct a set of words (or blocks of symbols) which is complete, meaning that probabilities of occurence of words from this set can be expressed as linear combinations of probabilities of occurence of these words at the previous time step. One can then setup and solve a recursion for block probabilities. We demonstrate an example of such PCA, which can be viewed as a simple model of diffusion of information or spread of rumors. Expressions for the density of ones are obtained for this rule using the proposed method.