Sadhu Tridib, Dhar Deepak
Department of Theoretical Physics, Tata Institute of Fundamental Research, Mumbai 400005, India.
Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Feb;85(2 Pt 1):021107. doi: 10.1103/PhysRevE.85.021107. Epub 2012 Feb 6.
We study the patterns formed by adding N sand grains at a single site on an initial periodic background in the Abelian sandpile models, and relaxing the configuration. When the heights at all sites in the initial background are low enough, one gets patterns showing proportionate growth, with the diameter of the pattern formed growing as N(1/d) for large N, in d dimensions. On the other hand, if sites with maximum stable height in the starting configuration form an infinite cluster, we get avalanches that do not stop. In this paper we describe our unexpected finding of an interesting class of backgrounds in two dimensions that show an intermediate behavior: For any N, the avalanches are finite, but the diameter of the pattern increases as N(α), for large N, with 1/2<α≤1. Different values of α can be realized on different backgrounds, and the patterns still show proportionate growth. The noncompact nature of growth simplifies their analysis significantly. We characterize the asymptotic pattern exactly for one illustrative example with α=1.
我们研究在阿贝尔沙堆模型中，在初始周期性背景的单个位置添加(N)个沙粒并使构型弛豫后形成的模式。当初始背景中所有位置的高度足够低时，会得到显示出成比例增长的模式，对于大的(N)，在(d)维中形成的模式直径以(N^{(1/d)})的形式增长。另一方面，如果起始构型中具有最大稳定高度的位置形成一个无限簇，我们会得到不会停止的雪崩。在本文中，我们描述了我们在二维中意外发现的一类有趣的背景，它们表现出中间行为：对于任何(N)，雪崩是有限的，但对于大的(N)，模式直径以(N^{(\alpha)})的形式增加，其中(1/2 < \alpha \leq 1)。在不同的背景上可以实现不同的(\alpha)值，并且模式仍然显示出成比例增长。增长的非紧致性质显著简化了它们的分析。我们精确地刻画了一个(\alpha = 1)的说明性示例的渐近模式。
