The Minimum Spanning Tree Constant in Geometrical Probability and Under the Independent Model: A Unified Approach

Abstract

Given $n$ uniformly and independently distributed points in the $d$-dimensional cube of unit volume, it is well established that the length of the minimum spanning tree on these $n$ points is asymptotic to $\beta_{\mathrm{MST}}(d)n^{(d - 1)/d}$, where the constant $\beta_{\mathrm{MST}}(d)$ depends only on the dimension $d$. It has been a major open problem to determine the constant $\beta_{\mathrm{MST}}(d)$. In this paper we obtain an exact expression for the constant $\beta_{\mathrm{MST}}(d)$ on a torus as a series expansion. Truncating the expansion after a finite number of terms yields a sequence of lower bounds; the first five terms give a lower bound which is already very close to the empirically estimated value of the constant. Our proof technique unifies the derivation for the MST asymptotic behavior for the Euclidean and the independent model.