A quadratic separation inequality for spanning trees
Gefei Cai
AI-assisted manuscript, October 2026
Abstract
We prove a quadratic separation inequality for weighted spanning trees on finite undirected networks and give two applications. First, for every finite $n$-vertex transitive network, the expected distance in the uniform spanning tree (UST) between two independent uniform vertices is at least $\sqrt{n}-1$, resolving a conjecture in [Benjamini–Kozma, Comm. Math. Phys. 259 (2005), 257–286]. Second, we prove that the growth exponent of three-dimensional loop-erased random walk (LERW) is at least $3/2$.