Exceptional points of two-dimensional random walks at multiples of the cover time

We study exceptional sets of the local time continuous-time simple random walk in scaled-up (by N) versions $$D_N\subseteq {\mathbb {Z}}^2$$ bounded open domains $$D\subseteq {R}}^2$$ . Upon exit from $$D_N$$ , lands on a “boundary vertex” and then reenters through boundary edge next step. In parametrization by at we prove that, times corresponding to $$\theta $$ -multiple cover suitably defined $$\lambda -thick (i.e., heavily visited) -thin lightly points are, as $$N\rightarrow \infty distributed according Liouville Quantum Gravity $$Z^D_\lambda with parameter -times critical value. For <1$$ also set avoided vertices (a.k.a. late points) where is order unity are $$Z^D_{\sqrt{\theta }}$$ The structure described well, that pinned Discrete Gaussian Free Field for thick thin random-interlacement occupation-time field points. results demonstrate universality these extremal problems.