Phase transition for the late points of random walk

Let X be a simple random walk in Znd with d≥3 and let tcov be the expected time it takes for X to visit all vertices of the torus. In joint work with Prévost and Rodriguez we study the set Lα of points that have not been visited by time αtcov and prove that it exhibits a phase transition: there exists α∗ so that for all α>α∗ and all ϵ>0 there exists a coupling between Lα and two i.i.d. Bernoulli sets B± on the torus with parameters n−(a±ϵ)d with the property that B−⊆Lα⊆B+ with probability tending to 1 as n→∞. When α≤α∗, we prove that there is no such coupling.