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

• Louigi Addario-Berry ^[1] ; Nicolas Broutin ^[2] ; Christina Goldschmidt ^[3] ; Grégory Miermont ^[4]
  1. [1] McGill University

     McGill University

     Canadá

     
  2. [2] University of Oxford

     University of Oxford

     Oxford District, Reino Unido

     
  3. [3] École Normale Supérieure de Lyon

     École Normale Supérieure de Lyon

     Arrondissement de Lyon, Francia

     
  4. [4] Institut Universitaire de France

     Institut Universitaire de France

     París, Francia

     

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• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 45, Nº. 5, 2017, págs. 3075-3144
• Idioma: inglés
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  • Consider the minimum spanning tree (MST) of the complete graph with nn vertices, when edges are assigned independent random weights. Endow this tree with the graph distance renormalized by n1/3n1/3 and with the uniform measure on its vertices. We show that the resulting space converges in distribution as n→∞n→∞ to a random compact measured metric space in the Gromov–Hausdorff–Prokhorov topology. We additionally show that the limit is a random binary RR-tree and has Minkowski dimension 33 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 RR-graphs, which generalise RR-trees, and are of independent interest.