Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Balanced spanning trees of the 2-by-N grid

Makenzie Gallagher, Kristopher Tapp

TL;DR

This paper computes the exact probability that a uniformly random spanning tree of the $2$-by-$n$ grid is balanced (contains a cut edge dividing vertices equally) and derives its explicit limits as $n$ grows. Using a dual-graph framework and a loop-counting strategy, the authors express the balanced-tree count $S_n$ in terms of squared numbers of spanning trees $T_{m-i}$ and obtain closed-form asymptotics via the Raff recurrence for $T_n$. The main results provide exact formulas for $S_n/T_n$ in both odd and even cases and yield precise limits: $(3+\sqrt{3})/9$ (odd) and $(1+4\sqrt{3})/(6\sqrt{3})$ (even). The work also compares uniform spanning-tree (UST) and minimum spanning-tree (MST) distributions, revealing surprising behavior in the even case and suggesting several directions for future study, including higher dimensions and near-balanced trees.

Abstract

We obtain an exact formula for the probability that a uniformly random spanning tree of the $2$-by-$n$ square grid is ``balanced'' in the sense that it has an edge whose removal partitions its vertices into two sets of equal size. We compute the exact limit of this probability as $n\rightarrow\infty$.

Balanced spanning trees of the 2-by-N grid

TL;DR

This paper computes the exact probability that a uniformly random spanning tree of the -by- grid is balanced (contains a cut edge dividing vertices equally) and derives its explicit limits as grows. Using a dual-graph framework and a loop-counting strategy, the authors express the balanced-tree count in terms of squared numbers of spanning trees and obtain closed-form asymptotics via the Raff recurrence for . The main results provide exact formulas for in both odd and even cases and yield precise limits: (odd) and (even). The work also compares uniform spanning-tree (UST) and minimum spanning-tree (MST) distributions, revealing surprising behavior in the even case and suggesting several directions for future study, including higher dimensions and near-balanced trees.

Abstract

We obtain an exact formula for the probability that a uniformly random spanning tree of the -by- square grid is ``balanced'' in the sense that it has an edge whose removal partitions its vertices into two sets of equal size. We compute the exact limit of this probability as .

Paper Structure

This paper contains 6 sections, 2 theorems, 12 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related work
  3. Set up
  4. Counting balanced trees
  5. Proof of Theorem \ref{['T:main']}
  6. Conclusion and future directions

Key Result

Theorem 1.1

Figures (5)

  • Figure 1: $\mathcal{G}_{11}$ (in red) and $\mathcal{G}^*_{11}$ (in yellow).
  • Figure 2: A spanning tree $T$ of $\mathcal{G}_{11}$ (grey) and the corresponding spanning tree $T^*$ of $\mathcal{G}^*_{11}$ (brown).
  • Figure 3: Removing the highlighted edge of $T$ adds an edge to $T^*$, which create a loop at $v_\infty$.
  • Figure 4: Starting from the spanning tree $T$ of Figure \ref{['F:tree_and_dual']}, an alternative balanced tree with the same $\gamma$ can be obtained by modifying $T$ in the two yellow end blocks and selecting the balanced cut edge among the locations marked by stars.
  • Figure 5: The $n$ loops in $\mathcal{G}_{n}^*$ that divide the vertices of $\mathcal{G}_n$ equally are enumerated for $n=11$ (left) and $n=10$ (right).

Theorems & Definitions (4)

  • Theorem 1.1
  • Lemma 4.1
  • proof
  • proof : Proof of Theorem \ref{['T:main']}