On the Minimum Spanning Tree Distribution in Grids

TL;DR

This work analyzes the minimum spanning tree (MST) distribution on large grid graphs, establishing sharp exponential bounds on the probability of any given spanning tree under the MST process. By leveraging the Lyons-Peres explicit MST formula and a bipartite-tree framework, it connects decay rates to a limiting power series f associated with spanning-tree families, and proves a universal lower bound $Q^-(\mathcal{F}) \ge \frac{1}{e\cdot \overline{f}}$ for bounded, convergent, neighbor-independent families. The paper also develops the notion of approximate passing times, the geometric mean of scaled passing times, and applies these to concrete families (centipede, fractal, double spiral, uniform) to compare their MST-probability decay bases. Overall, it links combinatorial tree structure on grids to probabilistic decay rates, offering a framework to bound and potentially determine MST-probability baselines across families. This has implications for understanding MST behavior in large grids and related redistricting-inspired applications.

Abstract

We study the minimum spanning tree distribution on the space of spanning trees of the $n$-by-$n$ grid for large $n$. We establish bounds on the decay rates of the probability of the most and the least probable spanning trees as $n\rightarrow\infty$.

Figures (4)

• Figure 1: Three spanning tree families. For the double spiral and centipede family, only the tree in $G(7)$ is shown, but the pattern extends in the obvious way. For the fractal family, introduced in Alon, each quadrant of $F_{k+1}$ is a copy of $F_{k}$, and these copies are connected in the middle by a copy of $F_1$.
• Figure 2: The bipartite graph $\mathcal{G}$ associated to a spanning tree $T\subset G(3)$
• Figure 3: $\textnormal{Prob}_{\textnormal{MST}}$ versus avg-stretch for 2 special trees in $G(8)$ and 200 random trees (100 each from Kruskal's and Wilson's algorithm). Here $\textnormal{Prob}_{\textnormal{MST}}$ is approximated via Proposition \ref{['P:russ']} using a random sample of 10000 choices of $\alpha\in S(T)$.
• Figure 4: The bipartite graph of the centipede $T_4\subset G(4)$.

Theorems & Definitions (35)

• Theorem 1.1
• Definition 4.1
• Proposition 4.2: Prop 5.1 of Russ; see also Theorem 3.5 of Models
• proof : Sketch of Proof
• Example 4.3
• Proposition 4.4: Theorem 3.4 of Models
• proof : Proof of Theorem \ref{['T:bound2']}
• Definition 6.1
• Definition 6.2
• Definition 6.3