To be more precise, the reduction amounts at most to a sublogarithmic factor. The key idea for the proof of Theorem \[subCrit\] is to analyze the connections in the graph $G(X^{(n)})$ at different length scales. Although at every scale the presence of long edges can be used to reduce chemical distances, still such shortcuts are sufficiently rare to yield only a sublogarithmic reduction factor in comparison to the linear growth rate $n$. Before going into the (technical) details, we provide a rough sketch of the proof. Put $p=(s+d)/(2s)$, and subdivide the torus $\T_n$ into $k_1=n^{(1-p)d}$ subcubes $Q_1,\ldots,Q_{k_1}$ of side length $n^p$. The probability that $(\xi,r)\in X^{(n)}$ with $r\ge n^p$ is of order at most $n^dn^{-sp}=n^{-(s-d)/2}$. On the other hand, if such a point does not exist, then the endpoints of any edge in $G(X^{(n)})$ are contained in adjacent subcubes. Hence, the number of subcubes that we need to visit if we move from $q(-n\mathsf{e}_1/4)$ to $q(n\mathsf{e}_1/4)$ is at least of order $n^{1-p}$.

Now, we continue by subdividing each of the subcubes $Q_1,\ldots,Q_{k_1}$ into subsubcubes of side length $n^{p^2}$. For each of the subcubes $Q_i$ the probability that $(\xi,r)\in X^{(n)}\cap Q_i$ with $r\ge n^{p^2}$ is of order at most $n^{pd}n^{-sp^2}=n^{-(s-d)p/2}$. The 