Annals of Probability

Localization in random geometric graphs with too many edges

Sourav Chatterjee and Matan Harel

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We consider a random geometric graph $G(\chi_{n},r_{n})$, given by connecting two vertices of a Poisson point process $\chi_{n}$ of intensity $n$ on the $d$-dimensional unit torus whenever their distance is smaller than the parameter $r_{n}$. The model is conditioned on the rare event that the number of edges observed, $|E|$, is greater than $(1+\delta )\mathbb{E}(|E|)$, for some fixed $\delta >0$. This article proves that upon conditioning, with high probability there exists a ball of diameter $r_{n}$ which contains a clique of at least $\sqrt{2\delta \mathbb{E}(|E|)}(1-\varepsilon )$ vertices, for any given $\varepsilon >0$. Intuitively, this region contains all the “excess” edges the graph is forced to contain by the conditioning event, up to lower order corrections. As a consequence of this result, we prove a large deviations principle for the upper tail of the edge count of the random geometric graph. The rate function of this large deviation principle turns out to be nonconvex.

Article information

Ann. Probab., Volume 48, Number 2 (2020), 574-621.

Received: November 2016
Revised: July 2019
First available in Project Euclid: 22 April 2020

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Zentralblatt MATH identifier

Primary: 60F10: Large deviations 05C80: Random graphs [See also 60B20] 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Random geometric graph Poisson point process large deviation localization


Chatterjee, Sourav; Harel, Matan. Localization in random geometric graphs with too many edges. Ann. Probab. 48 (2020), no. 2, 574--621. doi:10.1214/19-AOP1387.

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