Consider a random walk on the $d$ dimensional torus of size length $N$, $d>2$, and let $T$ be the first time all the vertices of the torus have been visited by this walk. For $0<\alpha<1$, the $\alpha$-late points are defined as the vertices not visited by this walk at time $\alpha T$. It is known that for $\alpha$ large enough the $\alpha$-late points are essentially i.i.d. on the torus, whereas this is not the case for $\alpha$ small enough. I will explain why this phase transition actually happens at some parameter $\alpha^\star>1/2$, which can be explicitly written in terms of the Green function on the $d$-dimensional infinite lattice. I will also describe the law of these $\alpha$-late points in the non i.i.d. region $1/2<\alpha<\alpha^\star$. Based on joint work with Pierre-François Rodriguez and Perla Sousi.