In the divisible sandpile model an initial mass distribution of i.i.d. random variables on an infinite vertex-transitive graph is redistributed by a local toppling rule.

We establish the least action principle, the abelian property and the conservation of density for this model. We show that a divisible sandpile with mean 1 and finite variance is almost surely not stabilizable.

We give quantitative estimates for the model on the discrete torus and relate the number of topplings to a discrete biLaplacian Gaussian field.

This is joint work with Lionel Levine, Mathav Murugan and Yuval Peres.