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Topology of a uniform spanning tree on a cylinder

Nikita Kalinin, Denis Rakhmankin

TL;DR

This work analyzes uniform spanning trees on cylindrical graphs $G_{n,m}=C_n\times P_m$ and reveals a rigid trunk-plus-short-branch geometry that persists as the cylinder length grows. Using Wilson's algorithm and loop-erased random walks, the authors establish exponential tails for branch lengths off the trunk and for the size of the LR-Slash, with constants depending only on the circumference $n$. The results hold both for the plain cylinder and the cylinder with a sink, providing a geometric mechanism that may explain the observed plateau-like avalanche distributions in Abelian sandpiles on cylinders via Dhar's burning correspondence. Numerics on small and large widths corroborate the exponential decay predictions and highlight the robustness of the trunk-dominated structure across parameters.

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 $θ=θ(n)\in(0,1)$, 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.

Topology of a uniform spanning tree on a cylinder

TL;DR

This work analyzes uniform spanning trees on cylindrical graphs and reveals a rigid trunk-plus-short-branch geometry that persists as the cylinder length grows. Using Wilson's algorithm and loop-erased random walks, the authors establish exponential tails for branch lengths off the trunk and for the size of the LR-Slash, with constants depending only on the circumference . The results hold both for the plain cylinder and the cylinder with a sink, providing a geometric mechanism that may explain the observed plateau-like avalanche distributions in Abelian sandpiles on cylinders via Dhar's burning correspondence. Numerics on small and large widths corroborate the exponential decay predictions and highlight the robustness of the trunk-dominated structure across parameters.

Abstract

We study uniform spanning trees (USTs) on the cylindrical graph . Fix a trunk 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 and , depending only on , such that for all and , 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.
Paper Structure (10 sections, 6 theorems, 33 equations, 8 figures, 1 algorithm)

This paper contains 10 sections, 6 theorems, 33 equations, 8 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related works.
  3. Notation and statements
  4. Graph with sink
  5. Numerical experiments
  6. Proofs
  7. Wilson's algorithm
  8. Proof of \ref{['thm:main']}
  9. Proof of \ref{['thm:sink']}
  10. Proof of the exponential estimate for the size of a Slash

Key Result

Theorem 1

Let $G_{n,m}$ be a cylindrical graph and let $T$ be a uniform spanning tree on $G_{n,m}$. There exists a trunk $L=L(T)\subset T$ such that, for every $l\in\mathbb{N}$, where $C>0$ and $\theta=\theta(n)\in(0,1)$ depend only on the circumference $n$ and are independent of $m$ and $l$.

Figures (8)

  • Figure 1: A spanning tree on the cylindrical graph $G_{4,12}$. The designated trunk is blue, the branch is red, and the remaining edges are black. The top and bottom boundaries are identified; that is, we glue together the edges cut according to the hatching.
  • Figure 2: An unwrapped view of the cylindrical graph $G_{4,15}^s$. The orange vertices are identified with the sink $s$. The red edges form an LR–Slash separating the left and right portions of the spanning tree.
  • Figure :
  • Figure : n=3, m=20000
  • Figure : n=3, m=20000
  • ...and 3 more figures

Theorems & Definitions (20)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1: Exponential tail for branches off some trunk of a UST on a cylinder
  • Theorem 2
  • Definition 5
  • Theorem 3
  • Definition 6
  • Definition 7
  • ...and 10 more