Generalized percolation games on the $2$-dimensional square lattice, and ergodicity of associated probabilistic cellular automata
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915679670632448 |
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| author | Bhasin, Dhruv Karmakar, Sayar Podder, Moumanti Roy, Souvik |
| author_facet | Bhasin, Dhruv Karmakar, Sayar Podder, Moumanti Roy, Souvik |
| 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_12199 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalized percolation games on the $2$-dimensional square lattice, and ergodicity of associated probabilistic cellular automata Bhasin, Dhruv Karmakar, Sayar Podder, Moumanti Roy, Souvik Probability 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. |
| title | Generalized percolation games on the $2$-dimensional square lattice, and ergodicity of associated probabilistic cellular automata |
| topic | Probability |
| url | https://arxiv.org/abs/2405.12199 |