  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 $\alpha=1$.

Statistical Mechanics