  We study the scaling limit of a divisible sandpile model associated to a
truncated $\alpha$-stable random walk. We prove that the limiting distribution
is related to an obstacle problem for a truncated fractional Laplacian. We also
provide, as a fundamental tool, precise asymptotic expansions for the
corresponding rescaled discrete Green's functions. In particular, the
convergence rate of these Green's functions to its continuous counterpart is
derived.
