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.

本文研究二维均匀生成树的热核。我们通过证明淬火热核的对角线部分的领先阶多项式行为发生对数对数波动来改进以前的工作。此外，我们给出了平均热核的两侧估计，并表明出现在热核淬火和平均版本的非对角线部分的指数是不同的。最后，我们推导出热核的各种缩放限制，其含义包括使我们能够锐化关于平均热核的对角线部分和相关简单随机游走的预期距离的已知渐近线。