Let X be a random walk on the torus of side length N in dimension \(d\geqslant 3\) with uniform starting point, and \(t_{\textrm{cov}}\) be the expected value of its cover time, which is the first time that X has visited every vertex of the torus at least once. For \(\alpha > 0\) , the set \(\mathcal {L}^{\alpha }\) of \(\alpha \) -late points consists of those points not visited by X at time \(\alpha t_{\textrm{cov}}\) . We prove the existence of a value \(\alpha _{{*}} \in (\frac{1}{2},1)\) across which \(\mathcal {L}^{\alpha }\) trivialises as follows: for all \(\alpha > \alpha _{{*}}\) and \(\varepsilon \geqslant N^{-c}\) there exists a coupling of \(\mathcal {L}^\alpha \) and two occupation sets \(\mathcal {B}^{\alpha _\pm }\) of i.i.d. Bernoulli fields having the same density as \(\mathcal {L}^{\alpha \pm \varepsilon }\) , which is asymptotic to \(N^{-(\alpha \pm \varepsilon )d}\) , with the property that the inclusion \( \mathcal {B}^{\alpha _+} \subseteq \mathcal {L}^{\alpha } \subseteq \mathcal {B}^{\alpha _-}\) holds with high probability as \(N \rightarrow \infty \) . On the contrary, when \(\alpha \leqslant \alpha _*\) there is no such coupling. Corresponding results also hold for the vacant set of random interlacements at high intensities. The transition at \(\alpha _{{*}}\) corresponds to the (dis-)appearance of ‘double-points’ (i.e. neighboring pairs of points) in \(\mathcal {L}^\alpha \) . We further describe the law of \(\mathcal {L}^{\alpha }\) for \(\alpha >\frac{1}{2}\) by adding independent patterns to \(\mathcal {B}^{\alpha _{\pm }}\) . In dimensions \(d \geqslant 4\) these are exactly all two-point sets. When \(d=3\) one must also include all connected three-point sets, but no other.
