We bound the probability that a continuous time simple random walk on the infinite percolation cluster on $Z^d$ returns to the origin at time $t$. We use this result to show that in dimensions 5 and higher the uniform spanning forest on infinite percolation clusters supported on graphs with infinitely many connected components a.s.
"Return Probabilities of a Simple Random Walk on Percolation Clusters." Electron. J. Probab. 10 250 - 302, 2005. https://doi.org/10.1214/EJP.v10-240