Entropy of Random Walk Range

We study the entropy of the set traced by an $n$-step random walk on ${\mathbb{Z}}^d$. We show that for $d\ge 3$, the entropy is of order $n$. For $d=2$, the entropy is of order $n\mathrm{/}{\log }^{2}n$. These values are essentially governed by the size of the boundary of the trace.