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September 2017 The scaling limit of the minimum spanning tree of the complete graph
Louigi Addario-Berry, Nicolas Broutin, Christina Goldschmidt, Grégory Miermont
Ann. Probab. 45(5): 3075-3144 (September 2017). DOI: 10.1214/16-AOP1132

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.

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Louigi Addario-Berry. Nicolas Broutin. Christina Goldschmidt. Grégory Miermont. "The scaling limit of the minimum spanning tree of the complete graph." Ann. Probab. 45 (5) 3075 - 3144, September 2017. https://doi.org/10.1214/16-AOP1132

Information

Received: 1 January 2015; Revised: 1 April 2016; Published: September 2017
First available in Project Euclid: 23 September 2017

zbMATH: 06812201
MathSciNet: MR3706739
Digital Object Identifier: 10.1214/16-AOP1132

Subjects:
Primary: 60C05
Secondary: 60F05

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.45 • No. 5 • September 2017
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