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

TL;DR

This paper proves the existence of a non-uniqueness phase for infinite clusters in hyperbolic space random connection models using spherical transform techniques, providing new bounds and insights into critical phenomena.

Contribution

It introduces a spherical transform approach to analyze non-uniqueness phases in hyperbolic random connection models, extending understanding beyond previous methods.

Findings

Non-uniqueness phase established in hyperbolic space models.

Spherical transform used to bound operator norms and analyze critical behavior.

Results apply to Boolean and weight-dependent hyperbolic models.

Abstract

A non-uniqueness phase for infinite clusters is proven for a class of marked random connection models on the $d$-dimensional hyperbolic space, ${\mathbb{H}^d}$, in a high volume-scaling regime. The approach taken in this paper utilizes the spherical transform on ${\mathbb{H}^d}$ to diagonalize convolution by the adjacency function and the two-point function and bound their $L^2\to L^2$ operator norms. Under some circumstances, this spherical transform approach also provides bounds on the triangle diagram that allows for a derivation of certain mean-field critical exponents. In particular, the results are applied to some Boolean and weight-dependent hyperbolic random connection models. While most of the paper is concerned with the high volume-scaling regime, the existence of the non-uniqueness phase is also proven without this scaling for some random connection models whose resulting graphs are almost surely not locally finite.