  We prove that the Abelian sandpile model on a random binary and binomial
tree, as introduced in \cite{rrs}, is not critical for all branching
probabilities $p<1$; by estimating the tail of the annealed survival time of a
random walk on the binary tree with randomly placed traps, we obtain some more
information about the exponential tail of the avalanche radius. Next we study
the sandpile model on $\mathbb{Z}^d$ with some additional dissipative sites: we
provide examples and sufficient conditions for non-criticality; we also make a
connection with the parabolic Anderson model. Finally we initiate the study of
the sandpile model with both sources and sinks and give a sufficient condition
for non-criticality in the presence of a finite number of sources, using a
connection with the homogeneous pinning model.
