  We study the entropy of the distribution of the set R_n of vertices visited
by a simple random walk on a graph with bounded degrees in its first n steps.
It is shown that this quantity grows linearly in the expected size of R_n if
the graph is uniformly transient, and sublinearly in the expected size if the
graph is uniformly recurrent with subexponential volume growth. This in
particular answers a question asked by Benjamini, Kozma, Yadin and Yehudayoff
(arXiv:0903.3179v1).
