Arm exponent for the Gaussian free field on metric graphs in intermediate dimensions
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arXiv
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| Format: | Preprint |
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2023
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| author | Drewitz, Alexander Prévost, Alexis Rodriguez, Pierre-François |
| author_facet | Drewitz, Alexander Prévost, Alexis Rodriguez, Pierre-François |
| contents | We investigate the bond percolation model on transient weighted graphs ${G}$ induced by the excursion sets of the Gaussian free field on the corresponding metric graph. We assume that balls in ${G}$ have polynomial volume growth with growth exponent $α$ and that the Green's function for the random on ${G}$ exhibits a power law decay with exponent $ν$, in the regime $1\leq ν\leq \fracα{2}$. In particular, this includes the cases of ${G}=\mathbb{Z}^{3}$ for which $ν=1$, and ${G}= \mathbb{Z}^{4}$ for which $ν=\fracα{2}=2$. For all such graphs, we determine the leading-order asymptotic behavior for the critical one-arm probability, which we prove decays with distance $R$, like $R^{-\fracν{2}+o(1)}$. Our results are, in fact, more precise and yield logarithmic corrections when $ν>1$ as well as corrections of order $\log \log R$ when $ν=1$. We further obtain very sharp upper bounds on truncated two-point functions close to criticality, which are new when $ν>1$ and essentially optimal when $ν=1$. This extends previous results from arXiv:2101.05801 and arXiv:1807.11117. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_10030 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Arm exponent for the Gaussian free field on metric graphs in intermediate dimensions Drewitz, Alexander Prévost, Alexis Rodriguez, Pierre-François Probability Mathematical Physics 60K35, 60G15, 60J45, 82B43 We investigate the bond percolation model on transient weighted graphs ${G}$ induced by the excursion sets of the Gaussian free field on the corresponding metric graph. We assume that balls in ${G}$ have polynomial volume growth with growth exponent $α$ and that the Green's function for the random on ${G}$ exhibits a power law decay with exponent $ν$, in the regime $1\leq ν\leq \fracα{2}$. In particular, this includes the cases of ${G}=\mathbb{Z}^{3}$ for which $ν=1$, and ${G}= \mathbb{Z}^{4}$ for which $ν=\fracα{2}=2$. For all such graphs, we determine the leading-order asymptotic behavior for the critical one-arm probability, which we prove decays with distance $R$, like $R^{-\fracν{2}+o(1)}$. Our results are, in fact, more precise and yield logarithmic corrections when $ν>1$ as well as corrections of order $\log \log R$ when $ν=1$. We further obtain very sharp upper bounds on truncated two-point functions close to criticality, which are new when $ν>1$ and essentially optimal when $ν=1$. This extends previous results from arXiv:2101.05801 and arXiv:1807.11117. |
| title | Arm exponent for the Gaussian free field on metric graphs in intermediate dimensions |
| topic | Probability Mathematical Physics 60K35, 60G15, 60J45, 82B43 |
| url | https://arxiv.org/abs/2312.10030 |