Balanced spanning trees of the 2-by-N grid

TL;DR

This paper derives an exact formula for the probability that a random spanning tree of a 2-by-n grid is balanced, and computes its limit as n approaches infinity, providing insights into the structure of such trees.

Contribution

It introduces an exact formula for the probability of balanced spanning trees in 2-by-n grids and analyzes its asymptotic behavior.

Findings

Exact probability formula for balanced spanning trees

Limit of the probability as n approaches infinity

Insights into the structure of spanning trees in grid graphs

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 $n\rightarrow\infty$.