Pattern formation in fast-growing sandpiles

We study the patterns formed by adding $N$ sand-grains at a single site on an initial periodic background in the Abelian sandpile models, and relaxing the configuration. When the heights at all sites in the initial background are low enough, one gets patterns showing proportionate growth, with the diameter of the pattern formed growing as $N^{1/d}$ for large $N$, in $d$-dimensions. On the other hand, if sites with maximum stable height in the starting configuration form an infinite cluster, we get avalanches that do not stop. In this paper, we describe our unexpected finding of an interesting class of backgrounds in two dimensions, that show an intermediate behavior: For any $N$, the avalanches are finite, but the diameter of the pattern increases as $N^{\alpha}$, for large $N$, with $1/2 < \alpha \leq 1$. Different values of $\alpha$ can be realized on different backgrounds, and the patterns still show proportionate growth. The non-compact nature of growth simplifies their analysis significantly. We characterize the asymptotic pattern exactly for one illustrative example with $\alpha=1$.