Phase transition for the late points of random walk

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