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-pilemodels, in the scaling limit, can be expressed in terms of the probability of survival of asingle diffusing particle in a medium with absorbing boundaries and space-dependent jumprates. The scaling function for the probability distribution of residence times isnon-universal, and depends on the probability distribution according to which grains areadded at different sites. We determine this function exactly for the one-dimensionalsand-pile when grains are added randomly only at the ends. For sand-piles with grainsadded everywhere with equal probability, in any dimension, and of arbitrary shape, weprove that, in the scaling limit, the probability that the residence time is greater thant is , where is the average mass of the pile in the steady state. We also study finite size corrections tothis function.