  This work deals with the divisible sandpile model when an initial
configuration sampled from a heavy-tailed distribution. Extending results of
Levine et al. (2015) and Cipriani et al. (2016) we determine sufficient
conditions for stabilization and non-stabilization on infinite graphs. We
determine furthermore that the scaling limit of the odometer on the torus is an
$\alpha$-stable random distribution.
