Topology of a Uniform Spanning Tree on a Cylinder

Abstract

We study uniform spanning trees (USTs) on the discrete cylinder $G_{n,m}=C_n\times P_m$, in the regime where the circumference \(n\) is fixed and the length $m$ tends to infinity. Using Wilson's algorithm, with the initial root on one boundary ring and the first walk started from the other, we single out a \emph{trunk} $L\subset T$: the first loop-erased random-walk path connecting the two boundary rings. We prove that the tree-distance from any fixed vertex to this trunk has an exponential tail with constants independent of $m$. As a consequence, the longest branch attached to the trunk is at most logarithmic in the length of the cylinder with high probability. Our motivation comes from the Abelian sandpile model on cylinders, and in particular from the step-like, or ``ladder'', avalanche-size distributions observed numerically by Eckmann--Nagnibeda--Perriard in \cite{EckmannNagnibedaPerriard2023}. Via Dhar's burning algorithm, recurrent sandpile configurations correspond to spanning trees, suggesting that the geometry of a typical UST may influence how avalanches propagate along the cylinder. The branch estimates above, their wired analogue, and the exponential estimate for the interface separating vertices whose paths to the sink pass through opposite boundary rings are intended as a first step towards isolating geometric UST observables that may be relevant to these plateau phenomena.