Tail exponents of the three-dimensional uniform spanning tree and Abelian sandpile

Xinyi Li; Runsheng Liu; Daisuke Shiraishi

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arXiv · 2605.19419

Tail exponents of the three-dimensional uniform spanning tree and Abelian sandpile

Xinyi Li·Runsheng Liu·Daisuke Shiraishi

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Abstract

We study the local geometry of the three-dimensional uniform spanning tree and its connection with the Abelian sandpile model. We obtain sharp tail exponents, up to subpolynomial errors, for the past of the origin in the three-dimensional UST and for the $0$-tree of the $0$-wired uniform spanning forest. As a principal application, we prove the corresponding three-dimensional Abelian sandpile avalanche exponents: the avalanche-cluster radius has tail exponent $1$, while both the avalanche-cluster size and the total number of topplings have tail exponent $1/3$. These results identify the leading power-law behaviour of three-dimensional sandpile avalanches and improve previously known bounds.

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Xinyi Li, Runsheng Liu, Daisuke Shiraishi. 2026-05-19. Tail exponents of the three-dimensional uniform spanning tree and Abelian sandpile. https://arxiv.org/abs/2605.19419

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