An Example of Computation of the Density of Ones in Probabilistic Cellular Automata by Direct Recursion

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.

Orbits of Bernoulli Measures in Cellular Automata

“…This approach has been recently applied to a probabilistic CA rule defined by w(1|000) = 0, w(1|001) = α, w(1|010) = 1, w(1|011) = 1, (75) w(1|100) = β , w(1|101) = γ, w(1|110) = 1, w(1|111) = 1, and w(0|b) = 1 − w(1|b) for all b ∈ {0, 1} 3 , where α, β , γ ∈ [0, 1] are fixed parameters. This rule can be viewed as a generalized simple model for diffusion of innovations on one-dimensional lattice (Fukś, 2016a). The complete set for this rule consists of blocks 101, 1001, 100001, .…”

“…one obtains, assuming that the initial measure is µ p , P n (0) = E ((p β − 1) (p α − 1)) n + F (1 − γ) n if αβ p 2 − (α + β )p + γ = 0, (G + Hn)(1 − γ) n−1 if αβ p 2 − (α + β )p + γ = 0, (79) where E, F, G, H are constants depending on parameters α, β , γ and p (for detailed formulae, see Fukś, 2016a). For αβ p 2 − (α + β )p + γ = 0, this is an example of a linear-exponential convergence of P n (0) toward its limiting value, the only one known for a binary rule.…”

An Example of a Deterministic Cellular Automaton Exhibiting Linear-exponential Convergence to the Steady State

“…and we can clearly see the aforementioned linear-exponential convergence. Very recently, a probabilistic CA has been discovered [15] where the density of ones converges to its stationary value in a linear-exponential fashion, just like in the above example of a degenerate hyperbolic fixed point in a finite-dimensional dynamical systems. This probabilistic CA could be viewed as a simple model for diffusion of innovations, spread of rumors, or a similar process involving transport of information between neighbours.…”