Largest component and sharpness in continuum percolation

Niclas Küpper; Mathew D. Penrose

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arXiv · 2407.10715

Largest component and sharpness in continuum percolation

Niclas Küpper·Mathew D. Penrose

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Abstract

We investigate the behavior of large connected components in the Poisson Random Connection model in non-critical regimes with any bounded connection function. We show that the asymptotic size of the largest component restricted to a window grows logarithmically in the volume of that window in the subcritical case, and linearly in the supercritical case. We also prove a sharpness result saying that the order of the cluster at the origin has an exponentially decaying tail in the subcritical regime.

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Niclas Küpper, Mathew D. Penrose. 2024-07-15. Largest component and sharpness in continuum percolation. https://arxiv.org/abs/2407.10715

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