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March 2020 Localization in random geometric graphs with too many edges
Sourav Chatterjee, Matan Harel
Ann. Probab. 48(2): 574-621 (March 2020). DOI: 10.1214/19-AOP1387

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.

Citation

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Sourav Chatterjee. Matan Harel. "Localization in random geometric graphs with too many edges." Ann. Probab. 48 (2) 574 - 621, March 2020. https://doi.org/10.1214/19-AOP1387

Information

Received: 1 November 2016; Revised: 1 July 2019; Published: March 2020
First available in Project Euclid: 22 April 2020

zbMATH: 07199855
MathSciNet: MR4089488
Digital Object Identifier: 10.1214/19-AOP1387

Subjects:
Primary: 05C80, 60D05, 60F10

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.48 • No. 2 • March 2020
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