The chemical distance in random interlacements in the low-intensity regime

In $\mathbb{Z}^d$ with $d\ge 5$, we consider the time constant $\rho_u$ associated to the chemical distance in random interlacements at low intensity $u \ll 1$. We prove an upper bound of order $u^{-1/2}$ and a lower bound of order $u^{-1/2+\varepsilon}$. The upper bound agrees with the conjectured scale in which $u^{1/2}\rho_u$ converges to a constant multiple of the Euclidean norm, as $u\to 0$. Along the proof, we obtain a local lower bound on the chemical distance between the boundaries of two concentric boxes, which might be of independent interest. For both upper and lower bounds, the paper employs probabilistic bounds holding as $u\to 0$; these bounds can be relevant in future studies of the low-intensity geometry.