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

Abstract

Consider the minimum spanning tree (MST) of the complete graph with $n$ vertices, when edges are assigned independent random weights. Endow this tree with the graph distance renormalized by $n^{1/3}$ and with the uniform measure on its vertices. We show that the resulting space converges in distribution as $n\to\infty$ to a random compact measured metric space in the Gromov–Hausdorff–Prokhorov topology. We additionally show that the limit is a random binary $\mathbb{R}$-tree and has Minkowski dimension $3$ almost surely. In particular, its law is mutually singular with that of the Brownian continuum random tree or any rescaled version thereof. Our approach relies on a coupling between the MST problem and the Erdős–Rényi random graph. We exploit the explicit description of the scaling limit of the Erdős–Rényi random graph in the so-called critical window, established in [Probab. Theory Related Fields 152 (2012) 367–406], and provide a similar description of the scaling limit for a “critical minimum spanning forest” contained within the MST. In order to accomplish this, we introduce the notion of $\mathbb{R}$-graphs, which generalise $\mathbb{R}$-trees, and are of independent interest.