The uniform spanning tree (UST) on $Zd$ was constructed by Pemantle
in 1991 as the limit of the UST on finite boxes $[-n,n]2$.
In this talk I will discuss the form of the heat kernel (i.e.
random walk transition probability) on this random graph.
I will compare the bounds for the UST with those obtained earlier
for supercritical percolation.
This is joint work with Takashi Kumagai and David Croydon.