Two-component abelian sandpile models

TL;DR

This paper explores two-component abelian sandpile models, revealing that models with two conservation laws only produce trivial avalanches due to nonlinear relations among toppling probabilities.

Contribution

It derives nonlinear relations for two-component quadratic abelian algebras and shows that models with two conservation laws are limited to trivial avalanche behavior.

Findings

Two-component models have nonlinear relations among toppling probabilities.

Models with two conservation laws produce only trivial avalanches.

Derived relations constrain the behavior of multi-component abelian sandpile models.

Abstract

In one-component abelian sandpile models, the toppling probabilities are independent quantities. This is not the case in multi-component models. The condition of associativity of the underlying abelian algebras impose nonlinear relations among the toppling probabilities. These relations are derived for the case of two-component quadratic abelian algebras. We show that abelian sandpile models with two conservation laws have only trivial avalanches.