Minimal spanning trees and Stein's method

Kesten and Lee proved that the total length of a minimal spanning tree on certain random point configurations in $\R^d$ satisfy a central limit theorem. They also raised the question: how to make these results quantitative? However, unlike other functionals studied in geometric probability, the problem of determining the convergence rate in the central limit theorem for Euclidean minimal spanning trees remained open. In this work, we establish bounds on the convergence rate for the Poissonized version of this problem. We also derive bounds on the convergence rate for the analogous problem in the setup of the lattice $\Z^d$. The main ingredients in the proof are (a) a quantification of the Burton-Keane argument for the uniqueness of the infinite open cluster, and (b) Stein's method of normal approximation.