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 tends to infinity, to a random measured metric space in the GromovHausdorffProkhorov topology. We additionally show that the limit is a random binary Rtree 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 ErdösRényi random graph. We exploit the explicit description of the scaling limit of the ErdösRényi random graph in the socalled critical window, established by the first three authors in an earlier paper, and provide a similar description of the scaling limit for a "critical minimum spanning forest" contained within the MST.
 Publication:

arXiv eprints
 Pub Date:
 January 2013
 arXiv:
 arXiv:1301.1664
 Bibcode:
 2013arXiv1301.1664A
 Keywords:

 Mathematics  Probability;
 Mathematics  Combinatorics;
 60C05 (Primary);
 60F05 (Secondary)
 EPrint:
 60 pages, 4 figures