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The size of components in continuum nearest-neighbor graphs. (English) Zbl 1111.60076

Summary: We study the size of connected components of random nearest-neighbor graphs with vertex set the points of a homogeneous Poisson point process in \(\mathbb R^d\). 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.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60D05 Geometric probability and stochastic geometry
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References:

[1] Häggström, O. and Meester, R. (1996). Nearest neighbour and hard sphere models in continuum percolation. Random Structures Algorithms 9 295–315. · Zbl 0866.60088 · doi:10.1002/(SICI)1098-2418(199610)9:3<295::AID-RSA3>3.0.CO;2-S
[2] Nanda, S. and Newman, C. M. (1999). Random nearest neighbor and influence graphs on \(\mathbfZ^d\). Random Structures Algorithms 15 262–278. · Zbl 0939.05075 · doi:10.1002/(SICI)1098-2418(199910/12)15:3/4<262::AID-RSA5>3.0.CO;2-7
[3] Zong, C. (1998). The kissing numbers of convex bodies—A brief survey. Bull. London Math. Soc. 30 1–10. · Zbl 0937.52014 · doi:10.1112/S0024609397003408
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