Critical Exponents for Marked Random Connection Models

2023/05/12 by Alejandro Caicedo, Caicedo, Alejandro, Matthew Dickson +1 · 3 citations
Mathematics · Physics and Astronomy · #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Markov Chains and Monte Carlo Methods

paper · pdf · doi:10.48550/arxiv.2305.07398

Abstract

Here we prove critical exponents for Random Connections Models (RCMs) with random marks. The vertices are given by a marked Poisson point process on ℝ^d and an edge exists between any pair of vertices independently with a probability depending upon their spatial displacement and on their respective marks. Given conditions on the edge probabilities, we prove mean-field lower bounds for the susceptibility and percolation functions. In particular, we prove the equality of the susceptibility and percolation critical intensities. If we assume that a form of the triangle condition holds, then we also prove that the susceptibility, percolation and cluster tail critical exponents exist and take their mean-field values. Our proof approach adapts the differential inequality and magnetization function approaches that have been previously applied to discrete homogeneous settings to our continuum marked setting. This includes a proof of the analyticity of the magnetization function in the required parameter regime.

Cited by

• On strong sharp phase transition in the random connection model
• Estimating the critical threshold for stretched-out hyperbolic random connection models
• Decay of connection probability in high-dimensional continuum percolation

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• Mean-field behaviour of the random connection model on hyperbolic space
• On strong sharp phase transition in the random connection model
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• Phase transitions and percolation at criticality in enhanced random connection models