Phase transition for the late points of random walk

Abstract Let X be a random walk on the torus of side length N in dimension $$d\geqslant 3$$ d ⩾ 3 with uniform starting point, and $$t_{\textrm{cov}}$$ t 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 $$\alpha > 0$$ α > 0 , the set $$\mathcal {L}^{\alpha }$$ L α of $$\alpha $$ α -late points consists of those points not visited by X at time $$\alpha t_{\textrm{cov}}$$ α t cov . We prove the existence of a value $$\alpha _{{*}} \in (\frac{1}{2},1)$$ α ∗ ∈ ( 1 2 , 1 ) across which $$\mathcal {L}^{\alpha }$$ L α trivialises as follows: for all $$\alpha > \alpha _{{*}}$$ α > α ∗ and $$\varepsilon \geqslant N^{-c}$$ ε ⩾ N - c there exists a coupling of $$\mathcal {L}^\alpha $$ L α and two occupation sets $$\mathcal {B}^{\alpha _\pm }$$ B α ± of i.i.d. Bernoulli fields having the same density as $$\mathcal {L}^{\alpha \pm \varepsilon }$$ L