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Corner angles of sandpile aggregates on a background of holes

Year: 2026
Type: preprint
Abstract: Start the abelian sandpile on $\mathbb Z^2$ with $n$ chips at the origin, on a background in which every site absorbs the first $H$ chips it receives (for $H<0$, every site starts with $-H$ chips), and let $S_{n,H}$ be the set of sites that fire. Holroyd, Levine, Mészáros, Peres, Propp and Wilson asked whether the limiting shape of $S_{n,H}$ is a regular $(4H+12)$-gon. We show that for $H\ge-1$ every corner of a subsequential limit of $S_{n,H}$, under any rescaling, has an interior angle $\alpha$ with $\alpha>\pi/2$ and $\sin\alpha\le1/(H+3)$. The corners of a regular $(4H+12)$-gon are sharper than this, so no such limit is a regular $(4H+12)$-gon; since $S_{n,-2}$ is a square for every $n$, this answers the question for every $H\ge-2$. A limit that is a convex polygon has a number of vertices that is divisible by $4$ and at least $2\pi/\arcsin(1/(H+3))$; it has at least $12$ vertices for $H=-1$ and at least $20$ for $H=0$. The proof combines the least action principle, a reflection argument and a count of last firings with the compactness argument of Pegden and Smart, and ends with a corner lemma for the limiting equation $\Delta U=s$, in which $H+2\le s\le H+3$ on the limit set and $s=0$ outside it.
DOI: 10.5281/zenodo.24069934
Language: English
Open Access status: green
OpenAlex ID: W7225661916
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