Critical Exponents for Marked Random Connection Models

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 ${\mathbb{R}}^d$ 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.

arXiv comment59 pages, 2 figures. Typos corrected, minor comments and references added all along the paper. The proofs of Lemma 2.7 and 3.10 have more details now. The proof of Lemma 3.8 is now done in finite volume to justify the use of Margulis-Russo inequality. Lemma 3.5 has been added to justify that the finite volume quantities involved converge to the appropiate ones in infinite volume

References in corpus (5)

• The phase transition in inhomogeneous random graphs
• The Alexander-Orbach conjecture holds in high dimensions
• Subcritical regimes in the Poisson Boolean model of continuum percolation
• The Triangle Condition for the Marked Random Connection Model
• Existence of subcritical percolation phases for generalised weight-dependent random connection models

Cited by in corpus (1)

• Non-Uniqueness Phase in Hyperbolic Marked Random Connection Models using the Spherical Transform