  Given a sequence of lattice approximations $D_N\subset\mathbb Z^2$ of a
bounded continuum domain $D\subset\mathbb R^2$ with the vertices outside $D_N$
fused together into one boundary vertex $\varrho$, we consider discrete-time
simple random walks in $D_N\cup\{\varrho\}$ run for a time proportional to the
expected cover time and describe the scaling limit of the exceptional level
sets of the thick, thin, light and avoided points. We show that these are
distributed, up a spatially-dependent log-normal factor, as the zero-average
Liouville Quantum Gravity measures in $D$. The limit law of the local time
configuration at, and nearby, the exceptional points is determined as well. The
results extend earlier work by the first two authors who analyzed the
continuous-time problem in the parametrization by the local time at $\varrho$.
A novel uniqueness result concerning divisible random measures and, in
particular, Gaussian Multiplicative Chaos, is derived as part of the proofs.
