Topology of a uniform spanning tree on a cylinder

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.