Combinatorial games on Galton-Watson trees involving several-generation-jump moves
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866913231879012352 |
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| author | Podder, Moumanti Bhasin, Dhruv |
| author_facet | Podder, Moumanti Bhasin, Dhruv |
| contents | We study the $k$-jump normal and $k$-jump misère games on rooted Galton-Watson trees, expressing the probabilities of various outcomes of these games as specific fixed points of certain functions that depend on $k$ and the offspring distribution. We discuss results on phase transitions pertaining to draw probabilities when the offspring distribution is Poisson$(λ)$ (i.e. for which values of $λ$, the draw probability is strictly positive). We compare the probabilities of the various outcomes of the $2$-jump normal game with those of the $2$-jump misère game, and a similar comparison is drawn between the $2$-jump normal game and the $1$-jump normal game, under the Poisson regime. We describe the rate of decay of the probability that the first player loses the $2$-jump normal game as $λ\rightarrow \infty$. Finally, we discuss a sufficient condition for the average duration of the $k$-jump normal game to be finite. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_02124 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Combinatorial games on Galton-Watson trees involving several-generation-jump moves Podder, Moumanti Bhasin, Dhruv Probability Combinatorics We study the $k$-jump normal and $k$-jump misère games on rooted Galton-Watson trees, expressing the probabilities of various outcomes of these games as specific fixed points of certain functions that depend on $k$ and the offspring distribution. We discuss results on phase transitions pertaining to draw probabilities when the offspring distribution is Poisson$(λ)$ (i.e. for which values of $λ$, the draw probability is strictly positive). We compare the probabilities of the various outcomes of the $2$-jump normal game with those of the $2$-jump misère game, and a similar comparison is drawn between the $2$-jump normal game and the $1$-jump normal game, under the Poisson regime. We describe the rate of decay of the probability that the first player loses the $2$-jump normal game as $λ\rightarrow \infty$. Finally, we discuss a sufficient condition for the average duration of the $k$-jump normal game to be finite. |
| title | Combinatorial games on Galton-Watson trees involving several-generation-jump moves |
| topic | Probability Combinatorics |
| url | https://arxiv.org/abs/2205.02124 |