Abelian Sandpiles and the Harmonic Model

Abelian sandpiles and algebraic models

Motivated by the coincidence of topological entropies the connection between abelian sandpiles and harmonic models was established by K. Schmidt and E. Verbitskiy (2009). The dissipative sandpile models were shown to be symbolic representations of algebraic $Z^d$-actions of the harmonic models. Both models are determined by so-called simple sandpile polynomials. We extend this result to arbitrary sandpile polynomials. Moreover, we show that any sandpile model determined by a factor of a sandpile polynomial acts as an equal entropy cover of the corresponding algebraic model. For a special class…

Harmonic dynamics of the Abelian sandpile

The abelian sandpile serves as a model to study self-organized criticality, a phenomenon occurring in biological, physical and social processes. The identity of the abelian group is a fractal composed of self-similar patches, and its limit is subject of extensive collaborative research. Here, we analyze the evolution of the sandpile identity under harmonic fields of different orders. We show that this evolution corresponds to periodic cycles through the abelian group characterized by the smooth transformation and apparent conservation of the patches constituting the identity. The dynamics indu…

无限图的沙堆群

Sandpile group of infinite graphs

本文以$C$-网为汇点在$\mathbb Z^2$上定义沙堆动力学，构造回归状态群并证明其同构于离散$S^1$-值调和函数群，还分析挠性、Pontryagin对偶及射影极限。