One-arm Probabilities for Metric Graph Gaussian Free Fields below and at the Critical Dimension

For the critical level-set of the Gaussian free field on the metric graph of $\mathbb Z^d$, we consider the one-arm probability $\theta_d(N)$, i.e., the probability that the boundary of a box of side length $2N$ is connected to the center. We prove that $\theta_d(N)$ 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 $\theta_d(N) \asymp N^{-2}$ 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,\ \theta_d(N)=N^{-\frac{d}{2}+o(1)};\ \text{(2) for}\ d>d_c,\ \theta_d(N)=N^{-2+o(1)}. \end{equation*} Prior to our work, Drewitz, Pr\'evost 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 $\theta_{d_c}(N) = N^{-2+o(1)}$, which is now established. In addition, in a recent concurrent work, Drewitz, Pr\'evost and Rodriguez independently obtained the up-to-constant upper bound for $d=3$.