The Annals of Applied Probability
- Ann. Appl. Probab.
- Volume 6, Number 2 (1996), 466-494.
The RSW theorem for continuum percolation and the CLT for Euclidean minimal spanning trees
We prove a central limit theorem for the length of the minimal spanning tree of the set of sites of a Poisson process of intensity $\lambda$ in $[0, 1]^2$ as $\lambda \to \infty$. As observed previously by Ramey, the main difficulty is the dependency between the contributions to this length from different regions of $[0, 1]^2$; a percolation-theoretic result on circuits surrounding a fixed site can be used to control this dependency. We prove such a result via a continuum percolation version of the Russo-Seymour-Welsh theorem for occupied crossings of a rectangle. This RSW theorem also yields a variety of results for two-dimensional fixed-radius continuum percolation already well known for lattice models, including a finite-box criterion for percolation and absence of percolation at the critical point.
Ann. Appl. Probab., Volume 6, Number 2 (1996), 466-494.
First available in Project Euclid: 18 October 2002
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 90C27: Combinatorial optimization
Alexander, Kenneth S. The RSW theorem for continuum percolation and the CLT for Euclidean minimal spanning trees. Ann. Appl. Probab. 6 (1996), no. 2, 466--494. doi:10.1214/aoap/1034968140. https://projecteuclid.org/euclid.aoap/1034968140