The Annals of Probability

The size of components in continuum nearest-neighbor graphs

Iva Kozakova, Ronald Meester, and Seema Nanda

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Abstract

We study the size of connected components of random nearest-neighbor graphs with vertex set the points of a homogeneous Poisson point process in ℝd. The connectivity function is shown to decay superexponentially, and we identify the exact exponent. From this we also obtain the decay rate of the maximal number of points of a path through the origin. We define the generation number of a point in a component and establish its asymptotic distribution as the dimension d tends to infinity.

Article information

Source
Ann. Probab., Volume 34, Number 2 (2006), 528-538.

Dates
First available in Project Euclid: 9 May 2006

Permanent link to this document
https://projecteuclid.org/euclid.aop/1147179981

Digital Object Identifier
doi:10.1214/009117905000000729

Mathematical Reviews number (MathSciNet)
MR2223950

Zentralblatt MATH identifier
1111.60076

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60G55: Point processes 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Keywords
Continuum percolation Poisson process size of components nearest-neighbor connections random graphs

Citation

Kozakova, Iva; Meester, Ronald; Nanda, Seema. The size of components in continuum nearest-neighbor graphs. Ann. Probab. 34 (2006), no. 2, 528--538. doi:10.1214/009117905000000729. https://projecteuclid.org/euclid.aop/1147179981


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References

  • Häggström, O. and Meester, R. (1996). Nearest neighbour and hard sphere models in continuum percolation. Random Structures Algorithms 9 295--315.
  • Nanda, S. and Newman, C. M. (1999). Random nearest neighbor and influence graphs on $\mathbfZ^d$. Random Structures Algorithms 15 262--278.
  • Zong, C. (1998). The kissing numbers of convex bodies---A brief survey. Bull. London Math. Soc. 30 1--10.