Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Inefficiency of the block approximation in diploid Probabilistic Cellular Automata

Emilio N. M. Cirillo, Joram L. Vliem, Dirk Schuricht, Cristian Spitoni

TL;DR

This work studies a diploid probabilistic cellular automaton formed by mixing additive ECA rules $60$ and $102$, revealing an absorbing zero-density phase near $\lambda=1/2$ on finite periodic lattices. It provides a rigorous four-step absorption proof at $\lambda=1/2$ and corroborates the zero-density regime with Monte Carlo simulations, while showing that standard mean-field and finite-block approximations fail to detect this phase for computationally feasible block sizes. The central contribution is identifying a fundamental limitation of finite-block closures arising from additive and mirror symmetries, which impose global constraints not captured by local marginals. These results have implications for designing approximation schemes for symmetric, linear probabilistic cellular automata and emphasize the need for methods that incorporate global configuration structure.

Abstract

We study a probabilistic cellular automaton obtained as a mixture of the additive elementary rules 60 and 102. We prove that, for any finite periodic lattice and for mixing parameter $λ=1/2$, the system almost surely reaches the absorbing all-zero configuration in finitely many steps. In addition, Monte Carlo simulations indicate as well the presence of a zero-density stationary state in a finite interval around $λ=1/2$. Despite this absorbing behavior, both mean-field and block approximation schemes predict a stationary state with non-zero density. This failure, traced to the additive and mirror symmetries of the deterministic components, highlights a fundamental limitation of finite-block approximation in capturing the global dynamics of probabilistic cellular automata.

Inefficiency of the block approximation in diploid Probabilistic Cellular Automata

TL;DR

This work studies a diploid probabilistic cellular automaton formed by mixing additive ECA rules and , revealing an absorbing zero-density phase near on finite periodic lattices. It provides a rigorous four-step absorption proof at and corroborates the zero-density regime with Monte Carlo simulations, while showing that standard mean-field and finite-block approximations fail to detect this phase for computationally feasible block sizes. The central contribution is identifying a fundamental limitation of finite-block closures arising from additive and mirror symmetries, which impose global constraints not captured by local marginals. These results have implications for designing approximation schemes for symmetric, linear probabilistic cellular automata and emphasize the need for methods that incorporate global configuration structure.

Abstract

We study a probabilistic cellular automaton obtained as a mixture of the additive elementary rules 60 and 102. We prove that, for any finite periodic lattice and for mixing parameter , the system almost surely reaches the absorbing all-zero configuration in finitely many steps. In addition, Monte Carlo simulations indicate as well the presence of a zero-density stationary state in a finite interval around . Despite this absorbing behavior, both mean-field and block approximation schemes predict a stationary state with non-zero density. This failure, traced to the additive and mirror symmetries of the deterministic components, highlights a fundamental limitation of finite-block approximation in capturing the global dynamics of probabilistic cellular automata.
Paper Structure (6 sections, 6 theorems, 47 equations, 8 figures, 2 tables)

This paper contains 6 sections, 6 theorems, 47 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Mixture of ECA and block approximation
  3. A highly symmetric diploid
  4. Rigorous results for the balanced case
  5. Mean Field failure and block approximation inefficiency
  6. Conclusions

Key Result

Lemma 1

Consider the evolved configuration $\zeta^{t}$ at time $t$ and the site $i\in\mathbb{Z}_n$. If at least two of the sites $i-1,i,i+1$ belong to $D_t$, then, for every choice of the Bernoulli variables at time $t+1$, the triple $(\zeta^{t+1}_{i-1},\zeta^{t+1}_i,\zeta^{t+1}_{i+1})$ is admissible.

Figures (8)

  • Figure 1: The left and right panels report, respectively, the evolution up to time $150$ of the 60 and 102 on a $100$-cell periodic lattice; the central panel reports the diploid mixture with parameter $\lambda=1/2$. Unit squares represent cells in state one. In the initial configuration the state of all the cells is zero, but for the one at the center of the lattice. In the plot the time runs upwards.
  • Figure 2: Time evolution of the diploid with $400$ cells, mixture of rules 60 and 102, with parameter $\lambda=1/2$. The initial configuration is a realization of a Bernoulli distribution of cells with parameter $1/2$. The time evolves upwards (a) $t\in[0,300]$ (b) $t\in[301,600]$ (c) $t\in[601,900]$ (d) $t\in[901,1095]$. At time $t=1080$ the zero-density state is reached.
  • Figure 3: Density as function of the mixing parameter $\lambda$ computed via Monte Carlo simulations for a lattice with $n=10^5$. The thermalization and decorrelation times were $10^4$ and $10^2$ respectively.
  • Figure 4: Density as function of the mixing parameter $\lambda$ computed via MF, BA of different sizes, and Monte Carlo simulations. For the BA we report the solution found initializing the iterative equations with the $1/2$ Bernoulli measure.
  • Figure 5: Decay of the density as function of the size $M$ of the BA for different $\lambda$. The dashed lines denote the corresponding asymptotic densities of the Monte Carlo simulations. For large $\lambda$ the BA agrees with the simulations already at $M=13$, while it is off in the region of the zero-density absorbing state.
  • ...and 3 more figures

Theorems & Definitions (13)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 1
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Theorem 1
  • ...and 3 more