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.
Categories: math-ph math.MP