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An exactly solvable evaporation-deposition PCA with long-distance interactions

Arvind Ayyer, Moumanti Podder

Abstract

We consider a probabilistic cellular automaton (PCA) of evaporation-deposition on the one-dimensional lattice having $n$ sites with periodic boundary conditions, in which each site, during each epoch, can be in one of two states: $0$ and $1$. Fix a positive integer $m\geqslant 2$. There are two types of transitions at each discrete time, which are as follows: (i) the first site in every contiguous block of $m$ $0$s becomes a $1$ with probability $p_1$, and (ii) the first site in every contiguous block of $(m-1)$ $0$s followed immediately by a $1$ also becomes a $1$ with probability $(1-p_2)$. As in a PCA, all of these transitions occur simultaneously. We show that the resulting discrete-time Markov chain is ergodic, and we give an explicit formula for its limiting distribution, the partition function and the density. We also propose necessary and sufficient conditions for this Markov chain to be reversible. For $m=2$, we provide a fully analytical expression for the free energy of this model.

An exactly solvable evaporation-deposition PCA with long-distance interactions

Abstract

We consider a probabilistic cellular automaton (PCA) of evaporation-deposition on the one-dimensional lattice having sites with periodic boundary conditions, in which each site, during each epoch, can be in one of two states: and . Fix a positive integer . There are two types of transitions at each discrete time, which are as follows: (i) the first site in every contiguous block of s becomes a with probability , and (ii) the first site in every contiguous block of s followed immediately by a also becomes a with probability . As in a PCA, all of these transitions occur simultaneously. We show that the resulting discrete-time Markov chain is ergodic, and we give an explicit formula for its limiting distribution, the partition function and the density. We also propose necessary and sufficient conditions for this Markov chain to be reversible. For , we provide a fully analytical expression for the free energy of this model.
Paper Structure (12 sections, 18 theorems, 117 equations, 7 figures)

This paper contains 12 sections, 18 theorems, 117 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Description of the model
  3. Main results
  4. Stationary distribution
  5. The case of $\beta=0^{n}$
  6. The case of $\beta\neq 0^{n}$
  7. A decomposition of $\alpha_{[k+1,k+\ell]}$ when $\ell \geqslant (m-1)$
  8. The final expression for $\mathop{\mathrm{\mathbb{P}}}\nolimits[\alpha\rightarrow\beta_{[k+\ell]}]$
  9. Verification of the master equation
  10. Completion of the proof
  11. The partition function and density
  12. Neighbourhood size $m = 2$

Key Result

Proposition 3.1

For any $n \geqslant m \geqslant 2$ and $p_1, p_2 \in (0, 1)$, the $m$-NED is translation-invariant.

Figures (7)

  • Figure 1: A pictorial representation of the stochastic update rules of the $m$-NED.
  • Figure 2: All transitions for the $m$-NED with $n = 3$ and $m = 2$ (resp. $m = 3$) on the left (resp. right). The probabilities of the transitions have not been written down to avoid cluttering the figure.
  • Figure 3: A contour plot of the free energy.
  • Figure 4: The part, $\alpha_{[k+1,k+\ell]}$, of $\alpha$ that we henceforth focus on
  • Figure 5: An example decomposition of $\alpha_{[k+1,k+\ell]}$ as given by \ref{["alpha'_decomposed"]}
  • ...and 2 more figures

Theorems & Definitions (34)

  • Proposition 3.1
  • proof
  • Corollary 3.2
  • Theorem 3.3
  • Example 3.4
  • Corollary 3.5
  • proof
  • Theorem 3.6
  • Theorem 3.7
  • Proposition 3.8
  • ...and 24 more