part of the path connecting $q(-n\mathsf{e}_1/4)$ and $q(n\mathsf{e}_1/4)$ in any such subcube is a nearest-neighbor path on the level of subsubcubes, typically $n^{p-p^2}$ steps are needed to cross that subcube. If this was true $i\in\{1,\ldots,k_1\}$, then we would obtain a lower bound for the chemical distance between $q(-n\mathsf{e}_1/4)$ and $q(n\mathsf{e}_1/4)$ that is of order $n^{1-p^2}$ and continuing in this fashion, we would obtain in fact a lower bound that is linear in $n$. However, at each level we have to deal with a small loss, which leads to the sublogarithmic correction term in the final lower bound.

In order to make this argument precise, it is convenient to introduce some notation similar to that used in fractal percolation, see [@falcGrim]. In order to define precisely the iteration mentioned in the previous paragraph, we need to ensure that at each layer the number of subcubes is an integer. Therefore, we define $a_0=n$, and then, inductively, $a_k=a_{k-1}/\lfloor a_{k-1}/n^{p^k}\rfloor$. We also put $b_k=a_{k-1}/a_k$. Next, in order to determine the position of subcubes in the $k$th layer, we use the index set $$J_k=\{(i_1,\ldots, i_k)\in (\Z^d)^k:\, i_j\in\{0,\ldots, b_j-1\}^d\text{ for all }j\in\{1,\ldots,k\}\}.$$ For $k\ge1$ and $I=(i_1,\ldots,i_k)\in J_k$, we define the site $z_I=a_1i_1+\cdots+a_ki_k$ and the cube $$Q_{I}=(-n/2,\ldots,-n/2)+z_{I}+[0,1]^da_k,$$ 