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On a class of rational matrices and interpolating polynomials related to the discrete Laplace operator
Mathematical Physics · Submitted 9 May 2007
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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 $L>2$.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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 $\mathbb{Z}_2$. | background 0.99 | 0.16 | 0.03 | asserted 0.91 |
| s1 | Consider a rational $L\times L$ matrix $\mathcal{H}$, whose inner entries $\mathcal{H}_{ij}$ satisfy $\dlap\mathcal{H}_{ij}=0$. | background 0.53 | 0.29 | 0.05 | asserted 0.95 |
| s2 | The matrix $\mathcal{H}$ is thus the classical finite difference five-points approximation of the Laplace operator in two variables. | interpretation 0.50 | 0.14 | 0.05 | asserted 1.00 |
| s3 | 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 $L>2$. | result 0.68 | 0.94 | 0.86 | asserted 1.00 |
| s4 | This result proves a conjecture formulated in the context of deterministic fixed-energy sandpile models in statistical mechanics. | interpretation 0.79 | 0.71 | 0.32 | asserted 1.00 |
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Original 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.
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