Balanced spanning trees of the 2-by-N grid

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 $n\rightarrow\infty$.

💡 Research Summary

The paper investigates the probability that a uniformly random spanning tree of the 2‑by‑N grid graph, denoted Gₙ, is “balanced,” meaning there exists an edge whose removal splits the vertex set into two equal halves. This question is motivated by recent redistricting algorithms (ReCom, Forest ReCom, SMC) that repeatedly generate random spanning trees and then look for a balanced cut edge to produce district maps. While prior work (e.g., CLL V23, CPT24) showed that a polynomial fraction of spanning trees are balanced for large grids, no exact probabilities were known.

The authors first recall the classic recurrence for the total number of spanning trees of a 2‑by‑N grid, derived by Raff (2008):
 Tₙ₊₂ = 4 Tₙ₊₁ – Tₙ, with T₁ = 1, T₂ = 4.
Solving this linear recurrence yields a closed form
 T_k = a (r^k – r^{–k}), where a = 1/(2√3) and r = 2 + √3.

To count balanced trees, the paper introduces the dual graph Gₙ* and the vertex v_∞ corresponding to the unbounded face of Gₙ. For any spanning tree T of Gₙ, the edges not in T correspond to a unique cycle γ in Gₙ* that always passes through v_∞. A balanced cut edge in T is precisely an edge whose dual lies on a loop γ that encloses exactly n vertices of Gₙ (i.e., splits the original graph in half).

The authors enumerate all possible loops γ that achieve this “half‑area” condition. Up to horizontal symmetry, there are exactly n such loops. They label them γ₀,…,γ_m where m = ⌊n/2⌋, and describe the combinatorial parameters associated with each loop:

• length(γ_i) – the number of positions where the balanced cut edge can be placed (2i+3 for odd n, 2i+2 for even n).
• b_i – a symmetry multiplier (1 if the loop is symmetric, 2 otherwise).
• T_{2m−i} – the number of spanning trees that can be placed independently in the two “end blocks” (each a 2‑by‑3 subgrid) that remain after fixing γ_i.

Summing over all i gives an exact expression for the number Sₙ of balanced spanning trees:

• Odd case (n = 2m+1):
   Sₙ = (n + m – 1) ∑{i=0}^{m} (6 + 4i) · T{2m−i}.

• Even case (n = 2m):
   Sₙ = (n + 2) ∑{i=1}^{m} (4 + 4i) · T{2m−i}.

Dividing by Tₙ and using the closed form for T_k, the ratio Sₙ/Tₙ can be expressed as a finite sum involving powers of r. To obtain the asymptotic probabilities as n → ∞, the authors let m → ∞ and apply the Dominated Convergence Theorem. They introduce x = r – 2 = 7 – 4√3 (≈0.0718) and evaluate the resulting infinite series analytically. The limits are:

• Odd n (n = 2m+1):
   lim_{m→∞} S_{2m+1}/T_{2m+1} = (3 + √3)/9 ≈ 0.52578.

• Even n (n = 2m):
   lim_{m→∞} S_{2m}/T_{2m} = (1 + 4√3)/(6√3) ≈ 0.76289.

These constants are the exact probabilities that a uniformly random spanning tree of a large 2‑by‑N grid is balanced, confirming and sharpening earlier polynomial‑lower‑bound results.

Beyond the uniform spanning tree (UST) model, the paper also explores the minimal spanning tree (MST) distribution, where edge weights are assigned independently at random and Kruskal’s algorithm is used to select a minimum‑weight tree. For n ≤ 5 the authors enumerate all edge‑weight permutations to compute exact MST balanced probabilities; for larger n they estimate these probabilities via one million random permutations. The empirical data reveal a striking pattern: for even n, MST probabilities are consistently higher than UST probabilities (e.g., n = 4: MST ≈ 0.78730 vs. UST ≈ 0.78571), while for odd n the opposite trend holds (e.g., n = 3: MST = 4/7 ≈ 0.57143 vs. UST = 6/10 = 0.6). This contradicts the intuitive expectation that MSTs, being “more constrained,” would be less likely to contain a balanced edge. The authors note that the asymptotic limits of the MST probabilities remain an open question.

The paper concludes by outlining several natural extensions:

1. k‑balanced trees – where k−1 edges partition the graph into k equal‑size components.
2. Higher‑dimensional grids – especially the 3‑by‑N case.
3. Near‑balanced trees – allowing a small tolerance in the size difference of the two components.
4. Exact MST asymptotics – determining whether the observed superiority of MST over UST for even n persists in the limit.

Overall, the work provides the first exact formulas and limiting constants for the balanced‑tree probability on the simplest non‑trivial grid family, offering both a rigorous benchmark for redistricting algorithms and a springboard for deeper combinatorial investigations.