The Annals of Probability

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

Louigi Addario-Berry, Nicolas Broutin, Christina Goldschmidt, and Grégory Miermont

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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.

Article information

Source
Ann. Probab. Volume 45, Number 5 (2017), 3075-3144.

Dates
Received: January 2015
Revised: April 2016
First available in Project Euclid: 23 September 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1506132034

Digital Object Identifier
doi:10.1214/16-AOP1132

Subjects
Primary: 60C05: Combinatorial probability
Secondary: 60F05: Central limit and other weak theorems

Keywords
Minimum spanning tree scaling limit R-tree R-graph Gromov–Hausdorff–Prokhorov distance critical random graphs

Citation

Addario-Berry, Louigi; Broutin, Nicolas; Goldschmidt, Christina; Miermont, Grégory. The scaling limit of the minimum spanning tree of the complete graph. Ann. Probab. 45 (2017), no. 5, 3075--3144. doi:10.1214/16-AOP1132. https://projecteuclid.org/euclid.aop/1506132034


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