Fixed-point forms of the parallel symmetric sandpile model

This paper presents a generalization of the sandpile model, called the parallel symmetric sandpile model, which inherits the rule of the symmetric sandpile model and implements them in parallel. We prove that although the parallel model produces really less number of fixed points than that by the sequential model, the forms of fixed points of the two models are the same. Moreover, our proof is a constructive one, which gives a nearly shortest way to reach a given fixed point form.