Quenched and averaged tails of the heat kernel of the two-dimensional uniform spanning tree

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.

Automated summary

This article investigates the heat kernel of the two-dimensional uniform spanning tree, improving previous work by demonstrating log-logarithmic fluctuations and giving two-sided estimates for the on-diagonal part of the kernel. It also shows that the exponents in different parts of the quenched and averaged versions of the heat kernel differ, and derives various scaling limits which sharpen known asymptotics for the averaged heat kernel and expected distance traveled by a simple random walk.