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

Abstract

Let $\dlap$ be the discrete Laplace operator acting on functions(or rational matrices) $f:\mathbf{Q}_L\rightarrow\mathbb{Q}$,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 theclassical finite difference five-points approximation of theLaplace operator in two variables. We give a constructive proofthat $\mathcal{H}$ is the restriction to $\mathbf{Q}_L$ of adiscrete harmonic polynomial in two variables for any $L>2$. Thisresult proves a conjecture formulated in the context ofdeterministic fixed-energy sandpile models in statisticalmechanics.