Let $G=(V,E)$ be a random $r$-regular graph on $n$ nodes. Perform a random walk on $G$, starting from a node chosen according to the walk stationary distribution (i.e. chosen uar from $V$).
Claim. If $U \subset V$ and the walk is at vertex $u \in U$ at a certain moment in time then the chance that the walk will still be inside $U$ at the next step is $|U|/|V|$.
The claim is wrong (cause otherwise the expected cover time for random cubic graphs would be $|V| \times \log|V|$, and i know it is slightly larger than that) ... but why? What wrong assumption am I making? What is the exact chance of staying in a given set of nodes?