Chemical distance in graphs of polynomial growth
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912478501273600 |
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| author | Gorski, Christian Procaccia, Eviatar B. |
| author_facet | Gorski, Christian Procaccia, Eviatar B. |
| contents | We prove an Antal-Pisztora type theorem for transitive graphs of polynomial growth. That is, we show that if $G$ is a transitive graph of polynomial growth and $p > p_c(G)$, then for any two sites $x, y$ of $G$ which are connected by a $p$-open path, the chemical distance from $x$ to $y$ is at most a constant times the original graph distance, except with probability exponentially small in the distance from $x$ to $y$. We also prove a similar theorem for general Cayley graphs of finitely presented groups, for $p$ sufficiently close to 1. Lastly, we show that all time constants for the chemical distance on the infinite supercritical cluster of a transitive graph of polynomial growth are Lipschitz continuous as a function of $p$ away from $p_c$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_09120 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Chemical distance in graphs of polynomial growth Gorski, Christian Procaccia, Eviatar B. Probability We prove an Antal-Pisztora type theorem for transitive graphs of polynomial growth. That is, we show that if $G$ is a transitive graph of polynomial growth and $p > p_c(G)$, then for any two sites $x, y$ of $G$ which are connected by a $p$-open path, the chemical distance from $x$ to $y$ is at most a constant times the original graph distance, except with probability exponentially small in the distance from $x$ to $y$. We also prove a similar theorem for general Cayley graphs of finitely presented groups, for $p$ sufficiently close to 1. Lastly, we show that all time constants for the chemical distance on the infinite supercritical cluster of a transitive graph of polynomial growth are Lipschitz continuous as a function of $p$ away from $p_c$. |
| title | Chemical distance in graphs of polynomial growth |
| topic | Probability |
| url | https://arxiv.org/abs/2507.09120 |