  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.

Probability