On the behavior of random walk around heavy points
  Consider a symmetric aperiodic random walk in $Z^d$, $d\geq 3$. There are points (called heavy points) where the number of visits by the random walk is close to its maximum. We investigate the local times around these heavy points and show that they converge to a deterministic limit as the number of steps tends to infinity. 
Endre Cs\'aki, Ant\'onia F\"oldes, P\'al R\'ev\'esz