Sandpile models and random walkers on finite lattices

A study was conducted on the abelian sandpile models and random-walk models on finite square lattices with open boundaries. The avalanche size distribution of the sandpile models, as well as the distributions of the lengths of the random-walk paths were calculated using various methods. It was shown that, due to the conservation laws, the averages <n> of the avalanche size distributions of the deterministic and stochastic models were the same, and that they were both equal to the average length of the random-walk paths starting from random sites on the same lattice.