  We show that the probability distribution of the residence-times of sand
grains in sandpile models, in the scaling limit, can be expressed in terms of
the survival probability of a single diffusing particle in a medium with
absorbing boundaries and space-dependent jump rates. The scaling function for
the probability distribution of residence times is non-universal, and depends
on the probability distribution according to which grains are added at
different sites. We determine this function exactly for the 1-dimensional
sandpile when grains are added randomly only at the ends. For sandpiles with
grains are added everywhere with equal probability, in any dimension and of
arbitrary shape, we prove that, in the scaling limit, the probability that the
residence time greater than t is exp(-t/M), where M is the average mass of the
pile in the steady state. We also study finite-size corrections to this
function.
