The Lower Tail of the Random Minimum Spanning Tree

Consider a complete graph Kn where the edges have costs given by independent random variables, each distributed uniformly between 0 and 1. The cost of the minimum spanning tree in this graph is a random variable which has been the subject of much study. This note considers the large deviation probability of this random variable. Previous work has shown that the log-probability of deviation by ε is −Ω(n), and that for the log-probability of Z exceeding ζ(3) this bound is correct; log Pr[Z ≥ ζ(3) + ε] = −Θ(n). The purpose of this note is to provide a simple proof that the scaling of the lower tail is also linear, log Pr[Z ≤ ζ(3) − ε] = −Θ(n).