Small subgraphs in the trace of a random walk

We consider the combinatorial properties of the trace of a random walk on the complete graph and on the random graph $G(n,p)$. In particular, we study the appearance of a fixed subgraph in the trace. We prove that for a subgraph containing a cycle, the threshold for its appearance in the trace of a random walk of length $m$ is essentially equal to the threshold for its appearance in the random graph drawn from $G(n,m)$. In the case where the base graph is the complete graph, we show that a fixed forest appears in the trace typically much earlier than it appears in $G(n,m)$.