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

Abstract

Colour an element of Zd white if its coordinates are coprime and black otherwise. What does this colouring look like when seen from a “uniformly chosen” random point of Zd? More generally, label every element of Zd by its greatest common divisor: what do the labels look like around a “uniform” random point of Zd? 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.