Exponential decay for the random connection model using asymptotic transitivity

2025/09/02 by Frankie Higgs, Higgs, Frankie · 2 citations
Mathematics · Physics and Astronomy · #60K35 #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2509.02310

openalex publication_date 2025/09/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Largest Component and Sharpness in Subcritical Continuum Percolation
• Exponential decay of the volume for Bernoulli percolation: a proof via stochastic comparison
• Power-law bounds for critical long-range percolation below the\n upper-critical dimension
• Lace Expansion and Mean-Field Behavior for the Random Connection Model

Cited by

• On strong sharp phase transition in the random connection model
• Strict monotonicity of critical points in independent long-range percolation models

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