One-arm Probabilities for Metric Graph Gaussian Free Fields below and at the Critical Dimension
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929418032644096 |
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| author | Cai, Zhenhao Ding, Jian |
| author_facet | Cai, Zhenhao Ding, Jian |
| contents | For the critical level-set of the Gaussian free field on the metric graph of $\mathbb Z^d$, we consider the one-arm probability $θ_d(N)$, i.e., the probability that the boundary of a box of side length $2N$ is connected to the center. We prove that $θ_d(N)$ is $O(N^{-\frac{d}{2}+1})$ for $3\le d\le 5$, and is $N^{-2+o(1)}$ for $d=6$. Our upper bounds match the lower bounds in a previous work by Ding and Wirth up to a constant factor for $3\le d\le 5$, and match the exponent therein for $d=6$. Combined with our previous result that $θ_d(N) \asymp N^{-2}$ for $d>6$, this seems to present the first percolation model whose one-arm probabilities are essentially completely understood in all dimensions. In particular, these results fully confirm Werner's conjectures (2021) on the one-arm exponents:
\begin{equation*}
\text{(1) for}\ 3\le d<d_c=6,\ θ_d(N)=N^{-\frac{d}{2}+o(1)};\ \text{(2) for}\ d>d_c,\ θ_d(N)=N^{-2+o(1)}.
\end{equation*}
Prior to our work, Drewitz, Prévost and Rodriguez obtained upper bounds for $d\in \{3, 4\}$, which are very sharp although lose some diverging factors. In the same work, they conjectured that $θ_{d_c}(N) = N^{-2+o(1)}$, which is now established. In addition, in a recent concurrent work, Drewitz, Prévost and Rodriguez independently obtained the up-to-constant upper bound for $d=3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_02397 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | One-arm Probabilities for Metric Graph Gaussian Free Fields below and at the Critical Dimension Cai, Zhenhao Ding, Jian Probability For the critical level-set of the Gaussian free field on the metric graph of $\mathbb Z^d$, we consider the one-arm probability $θ_d(N)$, i.e., the probability that the boundary of a box of side length $2N$ is connected to the center. We prove that $θ_d(N)$ is $O(N^{-\frac{d}{2}+1})$ for $3\le d\le 5$, and is $N^{-2+o(1)}$ for $d=6$. Our upper bounds match the lower bounds in a previous work by Ding and Wirth up to a constant factor for $3\le d\le 5$, and match the exponent therein for $d=6$. Combined with our previous result that $θ_d(N) \asymp N^{-2}$ for $d>6$, this seems to present the first percolation model whose one-arm probabilities are essentially completely understood in all dimensions. In particular, these results fully confirm Werner's conjectures (2021) on the one-arm exponents: \begin{equation*} \text{(1) for}\ 3\le d<d_c=6,\ θ_d(N)=N^{-\frac{d}{2}+o(1)};\ \text{(2) for}\ d>d_c,\ θ_d(N)=N^{-2+o(1)}. \end{equation*} Prior to our work, Drewitz, Prévost and Rodriguez obtained upper bounds for $d\in \{3, 4\}$, which are very sharp although lose some diverging factors. In the same work, they conjectured that $θ_{d_c}(N) = N^{-2+o(1)}$, which is now established. In addition, in a recent concurrent work, Drewitz, Prévost and Rodriguez independently obtained the up-to-constant upper bound for $d=3$. |
| title | One-arm Probabilities for Metric Graph Gaussian Free Fields below and at the Critical Dimension |
| topic | Probability |
| url | https://arxiv.org/abs/2406.02397 |