  We consider the thick points of random walk, i.e. points where the local time
is a fraction of the maximum. In two dimensions, we answer a question of Dembo,
Peres, Rosen and Zeitouni and compute the number of thick points of planar
random walk, assuming that the increments are symmetric and have a finite
moment of order two. The proof provides a streamlined argument based on the
connection to the Gaussian free field and works in a very general setting
including isoradial graphs. In higher dimensions, we study the scaling limit of
the thick points. In particular, we show that the rescaled number of thick
points converges to a nondegenerate random variable and that the centred
maximum of the local times converges to a randomly shifted Gumbel distribution.
