A note on log-concave random graphs

TL;DR

This paper investigates the connectivity threshold and giant component existence in a class of dependent random graphs defined by uniform distributions on generalized Orlicz balls, leveraging negative correlation properties.

Contribution

It introduces a threshold for connectivity and proves the existence of a unique giant component in these specialized dependent random graphs.

Findings

Established a connectivity threshold for the graphs.

Proved the existence of a unique giant component.

Utilized negative correlation properties of the distributions.

Abstract

We establish a threshold for the connectivity of certain random graphs whose (dependent) edges are determined by the uniform distributions on generalized Orlicz balls, crucially using their negative correlation properties. We also show the existence of a unique giant component for such random graphs.