| Título: | Distribution of components in the k-nearest neighbour random geometric graph for k below the connectivity threshold |
| Autores: | Falgas-Ravry, Victor; Umeå Universitet |
| Fecha: | 2013-01-04 |
| Publicador: | Electronic journal of probability |
| Fuente: |
 |
| Tipo: | Peer-reviewed Article |
| Tema: |
Random geometric graphs 60K35 |
| Descripción: | Let $S_{n,k}$ denote the random geometric graph obtained by placing points inside a square of area $n$ according to a Poisson point process of intensity $1$ and joining each such point to the $k=k(n)$ points of the process nearest to it.In this paper we show that if $\mathbb{P}(S_{n,k} \textrm{ connected})>n^{-\gamma_1}$ then the probability that $S_{n,k}$ contains a pair of `small' components `close' to each other is $o(n^{-c_1})$ (in a precise sense of `small' and 'close'), for some absolute constants $\gamma_1>0$ and $c_1 >0$. This answers a question of Walters. (A similar result was independently obtained by Balister.)As an application of our result, we show that the distribution of the connected components of $S_{n,k}$ below the connectivity threshold is asymptotically Poisson. |
| Idioma: | Inglés |
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