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 Zᵈ, we consider the one-arm probability θd(N), i.e., the probability that the boundary of a box of side length $2N$ is connected to the center. We prove that θd(N) is O(N-d/2+1) for 3≤ d≤ 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≤ d≤ 5, and match the exponent therein for $d=6$. Combined with our previous result that θd(N) 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: {equation*} (1) for\ 3≤ d<d_c=6,\ θ_d(N)=N-d/2+o(1);\ (2) for\ d>d_c,\ θ_d(N)=N⁻²⁺ᵒ⁽¹⁾. {equation*} Prior to our work, Drewitz, Pr\'evost and Rodriguez obtained upper bounds for d∈ \3, 4\, which are very sharp although lose some diverging factors. In the same work, they conjectured that θdc(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 $d=3$.