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

TL;DR

This paper investigates the local structure and graphon convergence of a coprime-based coloring and GCD labeling of integer lattices, revealing their asymptotic properties and percolation behavior.

Contribution

It introduces a detailed analysis of the local limits and graphon convergence for coprime colorings and GCD profiles on integer lattices, extending understanding of their large-scale structure.

Findings

Established the local limit of the GCD profile around a typical point.

Proved convergence of the associated graphon.

Analyzed the percolation properties of the coprime coloring.

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" point of $\mathbb{Z}^d$? More generally, label every element of $\mathbb{Z}^d$ by its GCD: what do the labels look like around a "uniform" point of $\mathbb{Z}^d$? We answer these questions and generalisations of them, provide results of graphon convergence, as well as a "local/graphon" convergence. One can also investigate the percolative properties of the colouring under study.