Annals of Probability

Localization in random geometric graphs with too many edges

Sourav Chatterjee and Matan Harel

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

Permanent link to this document
https://projecteuclid.org/euclid.aop/1587542673

Digital Object Identifier
doi:10.1214/19-AOP1387

Mathematical Reviews number (MathSciNet)
MR4089488

Zentralblatt MATH identifier
07199855

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

Keywords
Random geometric graph Poisson point process large deviation localization

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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