  This article investigates the heat kernel of the two-dimensional uniform
spanning tree. We improve previous work by demonstrating the occurrence of
log-logarithmic fluctuations around the leading order polynomial behaviour for
the on-diagonal part of the quenched heat kernel. In addition we give two-sided
estimates for the averaged heat kernel, and we show that the exponents that
appear in the off-diagonal parts of the quenched and averaged versions of the
heat kernel differ. Finally, we derive various scaling limits for the heat
kernel, the implications of which include enabling us to sharpen the known
asymptotics regarding the on-diagonal part of the averaged heat kernel and the
expected distance travelled by the associated simple random walk.
