The Annals of Applied Probability

Minimal spanning trees and Stein’s method

Sourav Chatterjee and Sanchayan Sen

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Abstract

Kesten and Lee [Ann. Appl. Probab. 6 (1996) 495–527] proved that the total length of a minimal spanning tree on certain random point configurations in $\mathbb{R}^{d}$ satisfies a central limit theorem. They also raised the question: how to make these results quantitative? Error estimates in central limit theorems satisfied by many other standard functionals studied in geometric probability are known, but techniques employed to tackle the problem for those functionals do not apply directly to the minimal spanning tree. Thus, the problem of determining the convergence rate in the central limit theorem for Euclidean minimal spanning trees has remained open. In this work, we establish bounds on the convergence rate for the Poissonized version of this problem by using a variation of Stein’s method. We also derive bounds on the convergence rate for the analogous problem in the setup of the lattice $\mathbb{Z}^{d}$.

The contribution of this paper is twofold. First, we develop a general technique to compute convergence rates in central limit theorems satisfied by minimal spanning trees on sequences of weighted graphs, including minimal spanning trees on Poisson points inside a sequence of growing cubes. Second, we present a way of quantifying the Burton–Keane argument for the uniqueness of the infinite open cluster. The latter is interesting in its own right and based on a generalization of our technique, Duminil-Copin, Ioffe and Velenik [Ann. Probab. 44 (2016) 3335–3356] have recently obtained bounds on probability of two-arm events in a broad class of translation-invariant percolation models.

Article information

Source
Ann. Appl. Probab., Volume 27, Number 3 (2017), 1588-1645.

Dates
Received: March 2015
Revised: May 2016
First available in Project Euclid: 19 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1500451236

Digital Object Identifier
doi:10.1214/16-AAP1239

Mathematical Reviews number (MathSciNet)
MR3678480

Zentralblatt MATH identifier
1371.60035

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60F05: Central limit and other weak theorems 60B10: Convergence of probability measures

Keywords
Minimal spanning tree central limit theorem Stein’s method Burton–Keane argument two-arm event

Citation

Chatterjee, Sourav; Sen, Sanchayan. Minimal spanning trees and Stein’s method. Ann. Appl. Probab. 27 (2017), no. 3, 1588--1645. doi:10.1214/16-AAP1239. https://projecteuclid.org/euclid.aoap/1500451236


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