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Sandpile monomorphisms and limits. (English) Zbl 07514678

Summary: We introduce a tiling problem between bounded open convex polyforms \(\hat{P}\subset\mathbb{R}^2\) with colored directed edges. If there exists a tiling of the polyform \(\hat{P}_2\) by \(\hat{P}_1\), we construct a monomorphism from the sandpile group \(G_{\Gamma_1}=\mathbb{Z}^{\Gamma_1}/\Delta (\mathbb{Z}^{\Gamma_1})\) on \(\Gamma_1=\hat{P}_1\cap \mathbb{Z}^2\) to the one on \(\Gamma_2=\hat{P}_2\cap \mathbb{Z}^2\). We provide several examples of infinite series of such tilings converging to \(\mathbb{R}^2\), and thus define the limit of the sandpile group on the plane.

MSC:

20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups
60K35 Interacting random processes; statistical mechanics type models; percolation theory
47D07 Markov semigroups and applications to diffusion processes
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