Self-avoiding walk is ballistic on graphs with more than one end
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915726821949440 |
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| author | Lehner, Florian Lindorfer, Christian Panagiotis, Christoforos |
| author_facet | Lehner, Florian Lindorfer, Christian Panagiotis, Christoforos |
| contents | We prove that on any transitive graph $G$ with infinitely many ends, a self-avoiding walk of length $n$ is ballistic with extremely high probability, in the sense that there exist constants $c,t>0$ such that $\mathbb{P}_n(d_G(w_0,w_n)\geq cn)\geq 1-e^{-tn}$ for every $n\geq 1$. Furthermore, we show that the number of self-avoiding walks of length $n$ grows asymptotically like $μ_w^n$, in the sense that there exists $C>0$ such that $μ_w^n\leq c_n\leq Cμ_w^n$ for every $n\geq 1$. Our results extend more generally to quasi-transitive graphs with infinitely many ends, satisfying the additional technical property that there is a quasi-transitive group of automorphisms of $G$ which does not fix an end of $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_13121 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Self-avoiding walk is ballistic on graphs with more than one end Lehner, Florian Lindorfer, Christian Panagiotis, Christoforos Combinatorics Group Theory Probability We prove that on any transitive graph $G$ with infinitely many ends, a self-avoiding walk of length $n$ is ballistic with extremely high probability, in the sense that there exist constants $c,t>0$ such that $\mathbb{P}_n(d_G(w_0,w_n)\geq cn)\geq 1-e^{-tn}$ for every $n\geq 1$. Furthermore, we show that the number of self-avoiding walks of length $n$ grows asymptotically like $μ_w^n$, in the sense that there exists $C>0$ such that $μ_w^n\leq c_n\leq Cμ_w^n$ for every $n\geq 1$. Our results extend more generally to quasi-transitive graphs with infinitely many ends, satisfying the additional technical property that there is a quasi-transitive group of automorphisms of $G$ which does not fix an end of $G$. |
| title | Self-avoiding walk is ballistic on graphs with more than one end |
| topic | Combinatorics Group Theory Probability |
| url | https://arxiv.org/abs/2403.13121 |