This analysis identifies the percolation phase transition and critical exponents for the random connection model on hyperbolic space, suggesting agreement with mean-field theory.
We study the random connection model on hyperbolic space Hᵈ in dimension $d=2,3$. Vertices of the spatial random graph are given as a Poisson point process with intensity λ>0. Upon variation of λ there is a percolation phase transition: there exists a critical value λc>0 such that for λ<λc all clusters are finite, but infinite clusters exist for λ>λc. We identify certain critical exponents that characterize the clusters at (and near) λc, and show that they agree with the mean-field values for percolation. We derive the exponents through isoperimetric properties of critical percolation clusters rather than via a calculation of the triangle diagram.
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Dickson et al. (2025) studied this question.
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