The scaling limit of the minimum spanning tree of the complete graph

The minimum spanning tree. Consider the complete graph, Kn, on vertices labelled by {1, 2, . . . , n}. Put independent random weights on the edges which are uniformly distributed on [0, 1] and find the spanning tree Mn of smallest total weight; this is the so-called minimum spanning tree (MST). Now think of Mn as a metric space by taking the metric to be the graph distance divided by n. We also endow Mn with a probability measure by placing mass 1/n on each vertex. Our main result is the following theorem.