The sandpile identity element on an ellipse

TL;DR

This paper studies the sandpile group on elliptical lattice subsets, showing that as the lattice spacing decreases, the identity element's pattern dominates almost the entire area, revealing a predictable structure.

Contribution

It demonstrates the asymptotic dominance of a biperiodic pattern in the sandpile identity element on elliptical lattice subsets as the lattice spacing approaches zero.

Findings

The identity element mainly consists of a biperiodic pattern with some noise.

As lattice spacing decreases, the pattern's area fraction approaches 100%.

The pattern becomes more dominant in the limit of small lattice spacing.

Abstract

We consider certain elliptical subsets of the square lattice. The recurrent representative of the identity element of the sandpile group on this graph consists predominantly of a biperiodic pattern, along with some noise. We show that as the lattice spacing tends to 0, the fraction of the area taken up by the pattern in the identity element tends to 1.