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The Triangle Condition for the Marked Random Connection Model
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abstract
We investigate a spatial random graph model whose vertices are given as a marked Poisson process on $\mathbb{R}^d$. Edges are inserted between any pair of points independently with probability depending on the spatial displacement of the two endpoints and on their marks. Upon variation of the Poisson density, a percolation phase transition occurs under mild conditions: for low density there are finite connected components only, whilst for large density there is an infinite component almost surely. Our focus is on the transition between the low- and high-density phase, where the system is critical. We prove that if the dimension is high enough and the edge probability function satisfies certain conditions, then an infrared bound for the critical connection function is valid. This implies the triangle condition, and thus mean-field behaviour. We achieve this result through combining the recently established lace expansion for Poisson processes with spectral estimates.
Forward citations
Cited by 4 Pith papers
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Percolation in the marked stationary Random Connection Model for higher-dimensional simplicial complexes
A sharp phase transition for q-percolation is proven for the marked stationary random connection model on higher-dimensional simplicial complexes, unifying the Boolean, Vietoris-Rips, and Cech cases.
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Decay of connection probability in high-dimensional continuum percolation
At high dimension d, the critical two-point connection probability in the random connection model decays like |x|^{-(d-2)} with an explicit anisotropic prefactor.
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Central limit theorems for the Euler characteristic in the Random Connection Model for higher-dimensional simplicial complexes
For the new Random Connection Model of simplicial complexes, a generalized Euler characteristic is shown to satisfy quantitative central limit theorems in high-intensity and large-window limits.
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Betti numbers in the Random Connection Model for higher-dimensional simplicial complexes and the Boolean model
A central limit theorem is established for Betti numbers in the marked random connection model, and as a corollary for the Boolean model with general convex grains.
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