## Annals of Probability

### Localization in random geometric graphs with too many edges

#### Abstract

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

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

Dates
Revised: July 2019
First available in Project Euclid: 22 April 2020

https://projecteuclid.org/euclid.aop/1587542673

Digital Object Identifier
doi:10.1214/19-AOP1387

Mathematical Reviews number (MathSciNet)
MR4089488

Zentralblatt MATH identifier
07199855

#### Citation

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. https://projecteuclid.org/euclid.aop/1587542673

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