Probability distribution of residence times of grains in sand-pile models

Abstract

We show that the probability distribution of the residence times of sand grains in sand-pile models, in the scaling limit, can be expressed in terms of the probability of survival 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 one-dimensional sand-pile when grains are added randomly only at the ends. For sand-piles with grains 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 is greater than t is exp (-t/M bar) , where M bar is the average mass of the pile in the steady state. We also study finite size corrections to this function.