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 (preprint).
"Entropy of Random Walk Range on Uniformly Transient and on Uniformly Recurrent Graphs." Electron. J. Probab. 15 1143 - 1160, 2010. https://doi.org/10.1214/EJP.v15-787