Topology of a uniform spanning tree on a cylinder

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 $θ=θ(n)\in(0,1)$, depending only on $n$, such that for all $m\in\mathbb{N}$ and $l\geq 0$, $$ \mathbb{P}\left(\text{UST has a branch off the trunk }L ,\text{ of length }\geq l \right) \leq Cm(n-1)θ^{l}. $$ 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.

💡 Research Summary

The paper investigates the geometric structure of a uniformly random spanning tree (UST) on the cylindrical graph (G_{n,m}=C_n\times P_m), where (C_n) is a cycle of length (n) (the circumference) and (P_m) is a path of length (m) (the height). The motivation comes from the Abelian sandpile model (ASM) on cylinders, where numerical experiments have revealed a striking “ladder” or step‑like distribution of avalanche sizes: for a wide range of intermediate sizes the probability is almost constant, forming plateaus rather than a smooth power law. Since Dhar’s burning algorithm establishes a bijection between recurrent sandpile configurations and spanning trees (with a sink added), the authors conjecture that the typical geometry of a UST should control how avalanches propagate along the cylinder.

The authors first define a “trunk” (L) as a simple path in the spanning tree that intersects every horizontal ring (R_k) of the cylinder. A “branch” is any maximal simple path that shares exactly one endpoint with the trunk and is otherwise disjoint from it. The main result (Theorem 1) states that there exist constants (C>0) and (\theta\in(0,1)), depending only on the circumference (n), such that for every length (l\ge0) \