Topology of a uniform spanning tree on a cylinder

TL;DR

This paper investigates the structure of uniform spanning trees on cylindrical graphs, establishing exponential tail bounds for branch lengths from a designated trunk, with implications for understanding avalanche behaviors in sandpile models.

Contribution

It provides the first exponential tail bounds for branch lengths in USTs on cylinders, linking tree geometry to sandpile avalanche phenomena.

Findings

Exponential tail bound for branch lengths from the trunk

Quantitative relationship between tree structure and avalanche size distributions

Insights into the geometry influencing sandpile avalanche plateaus

Abstract

We study uniform spanning trees (USTs) on the cylindrical graph $G = C_n \times P_m$. Fix a trunk $L$ as a designated simple path in the tree connecting the two boundary rings of the cylinder. We prove an exponential tail bound for the length of branches emanating from the trunk: there exist constants $C>0$ and $\theta=\theta(n)\in(0,1)$, depending only on $n$, such that for all $m\in\mathbb{N}$ and $l\geq 0$, $$ \mathbb{P}\left(\text{UST has a branch off the trunk }L \,\text{ of length }\geq l \right) \leq Cm(n-1)\theta^{l}. $$ Our work is motivated by the Abelian sandpile model on cylinders and, in particular, by the step-like (ladder) avalanche size distributions observed numerically in [Eckmann--Nagnibeda--Perriard, Abelian sandpiles on cylinders]. Via Dhar's burning algorithm, recurrent sandpile configurations correspond to spanning trees, so the geometry of a typical UST should influence how avalanches propagate along the cylinder. The trunk-with-short-branches structure and slash estimates proved here are intended as a first step towards a geometric explanation of these plateau phenomena for sandpile avalanches.