Sparse stable configurations of hard discs in a box

TL;DR

This paper constructs stable configurations of overlapping discs in a square, demonstrating limitations of the Metropolis algorithm's irreducibility in certain parameter ranges, with implications for the hardcore model.

Contribution

It provides explicit stable configurations of overlapping discs of radius proportional to 1/n, establishing bounds on the Metropolis algorithm's irreducibility for the hardcore model.

Findings

Constructed stable disc configurations in a unit square.

Proved the non-irreducibility of the Metropolis algorithm in certain regimes.

Confirmed the optimality of the configuration size bounds.

Abstract

We construct stable configurations of n overlapping discs of radius r in a unit square, with r = O(1/n). By a result of Diaconis, Lebeau, and Michel, this result is best possible, up to a constant factor. A consequence is that the Metropolis algorithm, a well-studied Markov chain on the hardcore model, is not irreducible in this range of parameters.