The double-bubble problem on the square lattice

TL;DR

This paper studies the minimal-perimeter arrangements of two finite point sets on a square lattice, providing detailed geometric descriptions, optimal perimeter calculations, and shape characterizations as the sets grow large.

Contribution

It offers a comprehensive geometric analysis of the lattice double-bubble problem, including explicit perimeter formulas and shape limits for large sets.

Findings

Explicit formulas for minimal perimeters in certain regimes

Identification of the Wulff shape of minimisers at large scales

Sharp bounds on differences between non-unique minimisers

Abstract

We investigate minimal-perimeter configurations of two finite sets of points on the square lattice. This corresponds to a lattice version of the classical double-bubble problem. We give a detailed description of the fine geometry of minimisers and, in some parameter regime, we compute the optimal perimeter as a function of the size of the point sets. Moreover, we provide a sharp bound on the difference between two minimisers, which are generally not unique, and use it to rigorously identify their Wulff shape, as the size of the point sets scales up.