On a class of rational matrices and interpolating polynomials related to the discrete Laplace operator

TL;DR

This paper proves that certain rational matrices related to the discrete Laplace operator are restrictions of discrete harmonic polynomials, confirming a conjecture in statistical mechanics about fixed-energy sandpile models.

Contribution

It provides a constructive proof that rational matrices satisfying the discrete Laplace equation are restrictions of harmonic polynomials, confirming a conjecture in the field.

Findings

Rational matrices satisfying the discrete Laplace equation are restrictions of harmonic polynomials.

The proof is constructive, providing explicit methods.

Confirms a conjecture in deterministic fixed-energy sandpile models.

Abstract

Let $\dlap$ be the discrete Laplace operator acting on functions (or rational matrices) $f:\mathbf{Q}_L\to\mathbb{Q}$, where $\mathbf{Q}_L$ is the two dimensional lattice of size $L$ embedded in $\mathbb{Z}_2$. Consider a rational $L\times L$ matrix $\mathcal{H}$, whose inner entries $\mathcal{H}_{ij}$ satisfy $\dlap\mathcal{H}_{ij}=0$. The matrix $\mathcal{H}$ is thus the classical finite difference five-points approximation of the Laplace operator in two variables. We give a constructive proof that $\mathcal{H}$ is the restriction to $\mathbf{Q}_L$ of a discrete harmonic polynomial in two variables for any $L>2$. This result proves a conjecture formulated in the context of deterministic fixed-energy sandpile models in statistical mechanics.