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arXiv · 2509.02310

Exponential decay for the random connection model using asymptotic transitivity

Frankie Higgs
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Abstract

We prove that the probability the cluster of the origin in a subcritical Poisson random connection model (RCM) has size at least $n$ decays exponentially as $n$ increases, under minimal assumptions. We extend a recent method of Vanneuville (arXiv:2304.12110) from Bernoulli percolation on vertex-transitive graphs to the RCM. The key idea is that the subcritical RCM can be constructed by site percolation on a very high-intensity RCM. The latter RCM becomes ``almost vertex-transitive'' in a certain sense at very high intensities, which is a new method that we expect to be useful for other problems. We obtain the result for connection functions with unbounded support, a setting in which it was not previously known.

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Frankie Higgs. 2025-09-02. Exponential decay for the random connection model using asymptotic transitivity. https://arxiv.org/abs/2509.02310

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