Gaussian approximation in random minimal directed spanning trees

Chinmoy Bhattacharjee

Published 2021-05-01Version 1

We study the total length of rooted edges in a random minimal directed spanning tree - first introduced in \cite{BR04} - in the unit cube $[0,1]^d$ for $d \ge 3$. In the case of $d=2$, a Dickman limit was proved in \cite{PW04}. This raises the question, what happens in dimensions three and higher and is it possible to prove a quantitative limit theorem? In this paper, we prove a Gaussian central limit theorem for $d \ge 3$. Moreover, by utilizing recent results from \cite{BM21} originating in Stein's method, we provide presumably optimal non-asymptotic bounds on the Wasserstein and the Kolmogorov distances between the distributions of total length of rooted edges, suitably normalized, and a standard Gaussian random variable.