  In the single-source sandpile model, a number $N$ grains of sand are
positioned at a central vertex on the 2-dimensional grid $\mathbb{Z}^2$. We
study the stabilisation of this configuration for a stochastic sandpile model
based on a parameter $M \in \mathbb{N}$. In this model, if a vertex has at
least $4M$ grains of sand, it topples, sending $k$ grains of sand to each of
its four neighbours, where $k$ is drawn according to some random distribution
$\gamma$ with support $\{1,\cdots,M\}$. Topplings continue, a new random number
$k$ being drawn each time, until we reach a stable configuration where all the
vertices have less than $4M$ grains. This model is a slight variant on the one
introduced by Kim and Wang.
  We analyse the stabilisation process described above as $N$ tends to infinity
(for fixed $M$), for various probability distributions $\gamma$. We focus on
two global parameters of the system, referred to as radius and avalanche
numbers. The radius number is the greatest distance from the origin to which
grains are sent during the stabilisation, while the avalanche number is the
total number of topplings made. Our simulations suggest that both of these
numbers have fairly simple asymptotic behaviours as functions of $\gamma$, $N$
and $M$ as $N$ tends to infinity. We also provide a more detailed analysis in
the case where $\gamma$ is the binomial distribution with parameter $p$, in
particular when $p$ tends to $1$. We exhibit a phase transition in that regime
at the scale $p \sim 1/N$.
