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Pré-Publication, Document De Travail (Preprint/Prepublication) Année : 2023

On the first and second largest connected component in the percolated random geometric graph

Résumé

Given λ > 0, p ∈ [0, 1] and a Poisson Point Process Po(λ) in R 2 with intensity λ, we consider the random graph G = G(λ, p) with vertex set Po(λ) in which every pair of vertices at distance at most 1 forms an edge with probability p, independently of other pairs. Our study is motivated by the question of locality for bond percolation on the standard random geometric graph G(λ, 1). We show that for a large class of graphs converging locally to G(λ, 1) in a suitable sense, the corresponding critical percolation thresholds converge to the one of G(λ, 1). The proof is based on a finite volume criterion. In this direction, we strengthen recent results of Penrose by showing that the size of the largest component rescaled by n converges almost surely to a constant, and providing sharp bounds for the size of the second-largest component.
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Dates et versions

hal-03879244 , version 1 (30-11-2022)
hal-03879244 , version 2 (20-07-2023)

Identifiants

  • HAL Id : hal-03879244 , version 2

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Lyuben Lichev, Bas Lodewijks, Dieter Mitsche, Bruno Schapira. On the first and second largest connected component in the percolated random geometric graph. 2023. ⟨hal-03879244v2⟩
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