On coprime percolation, the visibility graphon, and the local limit of the GCD profile

Abstract

Colour an element of ${\mathbb{Z}}^d$ white if its coordinates are coprime and black otherwise. What does this colouring look like when seen from a “uniformly chosen” random point of ${\mathbb{Z}}^d$? More generally, label every element of ${\mathbb{Z}}^d$ by its greatest common divisor: what do the labels look like around a “uniform” random point of ${\mathbb{Z}}^d$? We answer these questions and generalisations of them, which includes a result a “local/graphon” convergence. We also investigate the percolative properties of the colouring under study.