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Generalized percolation games on the $2$-dimensional square lattice, and ergodicity of associated probabilistic cellular automata

Dhruv Bhasin, Sayar Karmakar, Moumanti Podder, Souvik Roy

TL;DR

This work generalizes percolation games on the 2D square lattice by allowing traps, targets, and open states on both vertices and edges, and connects draw probabilities to the ergodicity of associated probabilistic cellular automata (PCAs). It introduces two principal PCA families, G_{p,q,r} and E_{r',s'}, along with their envelopes, to model the game dynamics and recurrences, proving that draw probabilities vanish in several near-zero regimes by proving ergodicity via a weight-function (potential) method. A key contribution is establishing ergodicity for elementary PCAs in parameter regions where traditional criteria are ineffective, thereby deriving rigorous draw-zero results for three- and two-parameter percolation games. The paper also outlines a precise mapping between bond and generalized percolation formulations, and provides detailed, regime-specific weight-function constructions to support the ergodicity claims, which has implications for understanding phase transitions and adversarial percolation in lattice systems.

Abstract

Each vertex of the infinite $2$-dimensional square lattice graph is assigned, independently, a label that reads trap with probability $p$, target with probability $q$, and open with probability $(1-p-q)$, and each edge is assigned, independently, a label that reads trap with probability $r$ and open with probability $(1-r)$. A percolation game is played on this random board, wherein two players take turns to make moves, where a move involves relocating the token from where it is currently located, say $(x,y) \in \mathbb{Z}^{2}$, to one of $(x+1,y)$ and $(x,y+1)$. A player wins if she is able to move the token to a vertex labeled a target, or force her opponent to either move the token to a vertex labeled a trap or along an edge labeled a trap. We seek to find a regime, in terms of $p$, $q$ and $r$, in which the probability of this game resulting in a draw equals $0$. We consider special cases of this game, such as when each edge is assigned, independently, a label that reads trap with probability $r$, target with probability $s$, and open with probability $(1-r-s)$, but the vertices are left unlabeled. Various regimes of values of $r$ and $s$ are explored in which the probability of draw is guaranteed to be $0$. We show that the probability of draw in each such game equals $0$ if and only if a certain probabilistic cellular automaton (PCA) is ergodic, following which we implement the technique of weight functions to investigate the regimes in which said PCA is ergodic.

Generalized percolation games on the $2$-dimensional square lattice, and ergodicity of associated probabilistic cellular automata

TL;DR

This work generalizes percolation games on the 2D square lattice by allowing traps, targets, and open states on both vertices and edges, and connects draw probabilities to the ergodicity of associated probabilistic cellular automata (PCAs). It introduces two principal PCA families, G_{p,q,r} and E_{r',s'}, along with their envelopes, to model the game dynamics and recurrences, proving that draw probabilities vanish in several near-zero regimes by proving ergodicity via a weight-function (potential) method. A key contribution is establishing ergodicity for elementary PCAs in parameter regions where traditional criteria are ineffective, thereby deriving rigorous draw-zero results for three- and two-parameter percolation games. The paper also outlines a precise mapping between bond and generalized percolation formulations, and provides detailed, regime-specific weight-function constructions to support the ergodicity claims, which has implications for understanding phase transitions and adversarial percolation in lattice systems.

Abstract

Each vertex of the infinite -dimensional square lattice graph is assigned, independently, a label that reads trap with probability , target with probability , and open with probability , and each edge is assigned, independently, a label that reads trap with probability and open with probability . A percolation game is played on this random board, wherein two players take turns to make moves, where a move involves relocating the token from where it is currently located, say , to one of and . A player wins if she is able to move the token to a vertex labeled a target, or force her opponent to either move the token to a vertex labeled a trap or along an edge labeled a trap. We seek to find a regime, in terms of , and , in which the probability of this game resulting in a draw equals . We consider special cases of this game, such as when each edge is assigned, independently, a label that reads trap with probability , target with probability , and open with probability , but the vertices are left unlabeled. Various regimes of values of and are explored in which the probability of draw is guaranteed to be . We show that the probability of draw in each such game equals if and only if a certain probabilistic cellular automaton (PCA) is ergodic, following which we implement the technique of weight functions to investigate the regimes in which said PCA is ergodic.
Paper Structure (33 sections, 14 theorems, 186 equations, 6 figures)

This paper contains 33 sections, 14 theorems, 186 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The probabilistic cellular automata we study in this paper
  3. Why the three-parameter percolation game is an all-encompassing set-up
  4. Organization of the rest of the paper
  5. Formal description of our games and the main results
  6. Formal description of the probabilistic cellular automata we are concerned with
  7. A brief discussion on PCAs in general
  8. The specific PCAs we work with in this paper
  9. How the PCA $\mathop{\mathrm{\widehat{E}}}\nolimits_{r',s'}$ is obtained as a special case of the PCA $\mathop{\mathrm{\widehat{G}}}\nolimits_{p,q,r}$
  10. A major contribution of this paper in establishing ergodicity of elementary PCAs
  11. Motivations for studying generalized / bond percolation games, and a brief review of pertinent literature
  12. In oligopolistic competitions
  13. As an adversarial avatar of percolation
  14. A brief discussion on Maker-Breaker percolation games
  15. Motivation for studying the PCAs we consider in this paper
  16. ...and 18 more sections

Key Result

Theorem 1

There exist $\epsilon_{1}$, $\epsilon_{2}$ and $\epsilon_{3}$, in $(0,1)$, such that whenever $(p,q,r) \in \Theta$ with $p \leqslant \epsilon_{1}$, $q \leqslant \epsilon_{2}$ and $r \leqslant \epsilon_{3}$, the inequality in three_cond_universal holds, and $(p,q,r)$ satisfies the constraints describ and

Figures (6)

  • Figure 1: Regions covered by \ref{['three_cond_1_simplified']} and \ref{['three_cond_4_simplified']}
  • Figure 2: Regions covered by Theorem \ref{['thm:three-parameter']}
  • Figure 3: 2D plots for regions covered by Theorem \ref{['thm:three-parameter']}
  • Figure 4: Illustrating the stochastic update rules for the PCA $\mathop{\mathrm{\widehat{G}}}\nolimits_{p,q,r}$
  • Figure 5: Illustrating the stochastic update rules for the PCA $\mathop{\mathrm{\widehat{E}}}\nolimits_{r',s'}$
  • ...and 1 more figures

Theorems & Definitions (23)

  • Theorem 1
  • Remark 2.1
  • Corollary 2.1
  • proof
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • proof
  • ...and 13 more