A general approach to massive upper bound for two-point function with application to self-avoiding walk torus plateau
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915362674573312 |
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| author | Liu, Yucheng |
| author_facet | Liu, Yucheng |
| contents | We prove a sufficient condition for the two-point function of a statistical mechanical model on $\mathbb{Z}^d$, $d > 2$, to be bounded uniformly near a critical point by $|x|^{-(d-2)} \exp [ -c|x| / ξ]$, where $ξ$ is the correlation length. The condition is given in terms of a convolution equation satisfied by the two-point function, and we verify the condition for strictly self-avoiding walk in dimensions $d > 4$ using the lace expansion. As an example application, we use the uniform bound to study the self-avoiding walk on a $d$-dimensional discrete torus with $d > 4$, proving a ``plateau'' of the torus two-point function, a result previously obtained for weakly self-avoiding walk in dimensions $d > 4$ by Slade. Our method has the potential to be applied to other statistical mechanical models on $\mathbb{Z}^d$ or on the torus. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_17321 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A general approach to massive upper bound for two-point function with application to self-avoiding walk torus plateau Liu, Yucheng Probability Mathematical Physics 82B27, 82B41, 60K35 We prove a sufficient condition for the two-point function of a statistical mechanical model on $\mathbb{Z}^d$, $d > 2$, to be bounded uniformly near a critical point by $|x|^{-(d-2)} \exp [ -c|x| / ξ]$, where $ξ$ is the correlation length. The condition is given in terms of a convolution equation satisfied by the two-point function, and we verify the condition for strictly self-avoiding walk in dimensions $d > 4$ using the lace expansion. As an example application, we use the uniform bound to study the self-avoiding walk on a $d$-dimensional discrete torus with $d > 4$, proving a ``plateau'' of the torus two-point function, a result previously obtained for weakly self-avoiding walk in dimensions $d > 4$ by Slade. Our method has the potential to be applied to other statistical mechanical models on $\mathbb{Z}^d$ or on the torus. |
| title | A general approach to massive upper bound for two-point function with application to self-avoiding walk torus plateau |
| topic | Probability Mathematical Physics 82B27, 82B41, 60K35 |
| url | https://arxiv.org/abs/2310.17321 |