학술논문
A critical constant for the $k$-nearest-neighbour model.
Document Type
Journal
Author
Balister, Paul (1-MEMP) AMS Author Profile; Bollobás, Béla (4-CAMBT) AMS Author Profile; Sarkar, Amites (1-WWA) AMS Author Profile; Walters, Mark (4-LNDQM) AMS Author Profile
Source
Subject
60 Probability theory and stochastic processes -- 60G Stochastic processes
60G55Point processes
82Statistical mechanics, structure of matter -- 82B Equilibrium statistical mechanics
82B43Percolation
60G55
82
82B43
Language
English
Abstract
Summary: ``Let $\scr P$ be a Poisson process of intensity 1 in a square $S_n$ of area $n$. For a fixed integer $k$, join every point of $\scr P$ to its $k$ nearest neighbours, creating an undirected random geometric graph $G_{n,k}$. We prove that there exists a critical constant $c_{\rm crit}$ such that, for $cc_{\rm crit}$, $G_{n,\lfloor c\log n\rfloor}$ is connected with probability tending to 1 as $n\rightarrow\infty$. This answers a question posed in [P. N. Balister\ et al., Adv. in Appl. Probab. {\bf 37} (2005), no.~1, 1--24; MR2135151 (2006c:05124)].''