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

M. T. Barlow, D. A. Croydon*, T. Kumagai

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish
Pages (from-to)57-111
Number of pages55
JournalProbability Theory and Related Fields
Volume181
Issue number1-3
DOIs
Publication statusPublished - 2021 Nov
Externally publishedYes

Keywords

  • Heat kernel
  • Random walk
  • Uniform spanning tree

ASJC Scopus subject areas

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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