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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.12199 |
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Table of Contents:
- 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.