  The divisible sandpile starts with i.i.d. random variables ("masses") at the
vertices of an infinite, vertex-transitive graph, and redistributes mass by a
local toppling rule in an attempt to make all masses at most 1. The process
stabilizes almost surely if m<1 and it almost surely does not stabilize if m>1,
where $m$ is the mean mass per vertex. The main result of this paper is that in
the critical case m=1, if the initial masses have finite variance, then the
process almost surely does not stabilize. To give quantitative estimates on a
finite graph, we relate the number of topplings to a discrete biLaplacian
Gaussian field.
