Approaching Criticality via the Zero Dissipation Limit in the Abelian Avalanche Model
Approaching Criticality via the Zero Dissipation Limit in the Abelian Avalanche Model
复制标题
通过阿贝尔雪崩模型中的零耗散极限逼近临界点
DOI：
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
E. Saada
中科院分区：
文献类型：
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作者：
A. Járai;F. Redig;E. Saada
巴克、唐、维森费尔德和达尔等人提出了离散高度阿贝尔沙堆模型作为自组织临界性概念的一个例子。当模型被修改为允许颗粒在每次倾倒时消失时，它被称为体积耗散。我们提供了一个连续的高度版本的阿贝尔沙堆模型，称为阿贝尔雪崩模型，它允许任意少量的耗散发生在每一个倾倒的详细研究。我们证明了对于非零耗散，阿贝尔雪崩模型的平稳测度的无穷体积极限是存在的，并且可以通过加权生成树测度得到。我们表明，在整个非零耗散制度，该模型是不关键的，即，局部观测量的空间协方差呈指数衰减。然后，我们研究了零耗散极限，并证明了自组织临界模型恢复，无论是静态措施和动力学。我们得到严格的界限倾倒概率，并引入一个指数描述其缩放临界。我们严格地建立了这个指数的平均场值，$$d > 4$$d>4。
The discrete height abelian sandpile model was introduced by Bak, Tang, Wiesenfeld and Dhar as an example for the concept of self-organized criticality. When the model is modified to allow grains to disappear on each toppling, it is called bulk-dissipative. We provide a detailed study of a continuous height version of the abelian sandpile model, called the abelian avalanche model, which allows an arbitrarily small amount of dissipation to take place on every toppling. We prove that for non-zero dissipation, the infinite volume limit of the stationary measure of the abelian avalanche model exists and can be obtained via a weighted spanning tree measure. We show that in the whole non-zero dissipation regime, the model is not critical, i.e., spatial covariances of local observables decay exponentially. We then study the zero dissipation limit and prove that the self-organized critical model is recovered, both for the stationary measure and for the dynamics. We obtain rigorous bounds on toppling probabilities and introduce an exponent describing their scaling at criticality. We rigorously establish the mean-field value of this exponent for $$d > 4$$d>4.