Mathematical aspects of the abelian sandpile model
Mathematical aspects of the abelian sandpile model
复制标题
阿贝尔沙堆模型的数学方面
DOI：
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发表时间：
2005
期刊：
影响因子：
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通讯作者：
F. Redig
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文献类型：
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作者：
F. Redig
1988年，巴克、唐和维森费尔德（BTW）提出了一种他们称之为“自组织临界性”的晶格模型。自该模型出现以来，无论是在物理学还是在数学文献中，都对它进行了深入的研究。它显示了如何一个简单的动力学可以导致非常复杂的结构的出现，并驱动系统走向一个稳定的状态，其中共享几个属性的平衡系统在临界点，例如幂律衰减的集群大小和高度变量的相关性。几年后，Deepak达尔推广了这个模型，发现了其中的“加法算子的阿贝尔群结构”，并称之为“阿贝尔沙堆模型”（以下简称ASM）。他研究了平稳测度的自组织临界性质，并给出了循环配置的算法特征，即所谓的“燃烧算法”。该算法给出了ASM的递归配置和有根生成树之间的一一对应关系。与生成树的对应关系使普里兹耶夫能够在无限体积极限下计算二维的高度概率。某些特殊事件的概率-所谓的“弱允许簇”-可以使用“孟买技巧”在无限体积限制下精确计算。在物理学文献中，人们用标度变元、重整化群方法和共形场论（d = 2）研究了临界指数，并认为模型的上临界维数为d = 4 [35]。达尔和Majumdar研究了Bethe晶格上的模型，他们使用传递矩阵方法在热力学极限下精确计算了各种相关函数和雪崩团簇大小分布。自从发现了循环构形集合的阿贝尔群结构以来，在数学文献（特别是组合学和代数组合学）中，人们以“芯片发射博弈”、“狄利克雷博弈”、“狄利克雷博弈”等名称重新引入了这一模型。
In 1988, Bak, Tang and Wiesenfeld (BTW) introduced a lattice model of what they called “self-organized criticality”. Since its appearance, this model has been studied intensively, both in the physics and in the mathematics literature. It shows how a simple dynamics can lead to the emergence of very complex structures and drive the system towards a stationary state which shares several properties of equilibrium systems at the critical point, e.g. power-law decay of cluster sizes and of correlations of the height-variables. Some years later, Deepak Dhar generalized the model, discovered the “abelian group structure of addition operators” in it and called it “the abelian sandpile model”( abbreviated from now on ASM). He studied the self-organized critical nature of the stationary measure and gave an algorithmic characterization of recurrent configurations, the so-called “burning algorithm”. This algorithm gives a one-to one correspondence between the recurrent configurations of the ASM and rooted spanning trees. The correspondence with spanning trees allowed Priezzhev to compute the height probabilities in dimension 2 in the infinite-volume limit. Probabilities of certain special events -so-called “weakly allowed clusters”- can be computed exactly in the infinite-volume limit using the “Bombay-trick”. In the physics literature people studied critical exponents with scaling arguments, renormalization group method and conformal field theory (in d = 2), and it is argued that the upper critical dimension of the model is d = 4 [35]. Dhar and Majumdar studied the model on the Bethe lattice where they computed various correlation functions and avalanche cluster-size distributions exactly in the thermodynamic limit, using a transfer matrix approach. Since the discovery of the abelian group structure of the set of of recurrent configurations, in the mathematics literature (especially combinatorics and algebraic combinatorics) one (re)introduces the model under the names “chip-firing game, Dirichlet