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

TL;DR

This paper analyzes the tail exponents of the three-dimensional uniform spanning tree and Abelian sandpile model, providing precise power-law behaviors and improving previous bounds.

Contribution

It derives sharp tail exponents for the UST and Abelian sandpile model in three dimensions, revealing their leading power-law behaviors.

Findings

Tail exponent for the avalanche-cluster radius is 1.

Tail exponent for the avalanche-cluster size is 1/3.

Tail exponent for the total number of topplings is 1/3.

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.