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Geometric Random Graphs on Circles

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Advances in Probability and Mathematical Statistics

Part of the book series: Progress in Probability ((PRPR,volume 79))

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Abstract

Given a dense countable set in a metric space, the infinite random geometric graph is the random graph with the given vertex set and where any two points at distance less than 1 are connected, independently, with some fixed probability. It has been observed by Bonato and Janssen that in some, but not all, such settings, the resulting graph does not depend on the random choices, in the sense that it is almost surely isomorphic to a fixed graph. While this notion makes sense in the general context of metric spaces, previous work has been restricted to sets in Banach spaces. We study the case when the underlying metric space is a circle of circumference L, and find a surprising dependency of behaviour on the rationality of L.

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Acknowledgement

This work was supported in part by NSERC of Canada.

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Correspondence to Omer Angel .

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Angel, O., Spinka, Y. (2021). Geometric Random Graphs on Circles. In: Hernández‐Hernández, D., Leonardi, F., Mena, R.H., Pardo Millán, J.C. (eds) Advances in Probability and Mathematical Statistics. Progress in Probability, vol 79. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-85325-9_2

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