Spatial Asymmetric Two dimensional Continuous Abelian Sandpile Model
N. Azimi-Tafreshi, H. Dashti-Naserabadi, S. Moghimi-Araghi
我们在连续阿贝尔沙堆模型中引入了一些不对称性，如有向性和椭圆性。分析了不同高度的概率分布，并给出了相应的场论。此外，我们还找到了与一些高度变量相关的字段。
We insert some asymmetries in the continuous Abelian sandpile models, such as directedness and ellipticity. We analyze probability distribution of different heights and also find the field theory corresponding to the models. Also we find the fields associated with some height variables.
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We present a sandpile model, in which the instability of a site is determined also by the variables in a neighbourhood. This is a modification of the Abelian Sandpile Model, in which abelianity is preserved: it shares several mathematical properties of the original abelian model, while producing a more realistic dynamics. We show how our model presents interesting hydrodynamic features.
Two-component abelian sandpile models
在单分量阿贝尔沙堆模型中，倾倒概率是相互独立的量。在多组分模型中，情况并非如此。基础交换代数的结合性条件使得倾倒概率之间存在非线性关系。这些关系推导的情况下，两个组件的二次阿贝尔代数。我们证明了具有两个守恒律的阿贝尔沙堆模型只有平凡的雪崩。
We study critical properties of the continuous Abelian sandpile model with anisotropies in toppling rules that produce ordered patterns on it. Also we consider the continuous directed sandpile model perturbed by a weak quenched randomness and study critical behavior of the model using perturbative conformal field theory and show the model has a new random fixed point.
Abelian Sandpile Model: a Conformal Field Theory Point of View
本文导出了$c=-2$共形场论中与阿贝尔沙堆模型中弱允许团簇有关的标度场，并指出了这两种模型之间的直接关系。
On the Probability of Occurrence of Clusters in Abelian Sandpile Model
我们对正方格子上的阿贝尔沙堆模型（ASM）进行了大量的模拟。我们已经估计了许多集群的观测概率。有些是在良好的协议与以前的分析结果，而有些显示模拟和分析结果之间的差异。
Non conservative Abelian sandpile model with BTW toppling rule
研究了Tsuchiya and Katori，Phys. Rev. E { \bf 61}，1183（2000）]中引入BTW倾倒规则的非保守Abel沙堆模型.使用的模型和数值模拟中所涉及的不同的能量尺度的标度分析表明，该模型属于一个普遍性的类，从文献中考虑的以前的模型不同。
Continuous Abelian Sandpile Model in Two Dimensional Lattice
本文定义了一种新的沙堆模型，它与阿贝尔沙堆模型（ASM）非常相似，但高度变量是连续变量。利用我们在模型中定义的倾倒规则，我们证明了模型可以映射到ASM，因此两个模型的一般性质是相同的。然而，新的模型允许我们研究一些问题，如非常小的质量对高度概率的影响，不同的边界条件，等等。
The Directed Abelian Sandpile Model on Cylinders
研究定向圆柱形晶格上二维阿贝尔沙堆模型的周期边界条件与单侧耗散，揭示沙堆群的代数结构与驱动动力学周期性的联系。
Higher Order and boundary Scaling Fields in the Abelian Sandpile Model
阿贝尔沙堆模型（ASM）是与$c=-2$共形场论相关的一种自组织临界性（SOC）模型。以前曾提出过一些高度团簇对应的共形场。在这里，我们得出的第一个修正等领域，在场论的方法，当晶格参数是非零的，并考虑他们在边界的存在。
Rare events and breakdown of simple scaling in the Abelian sandpile
由于不变性和守恒性，二维阿贝尔沙堆服从多重分形，而不是有限尺寸标度。在热力学极限，一个消失的小部分大雪崩占主导地位的统计和一个恒定的间隙缩放恢复在较高的时刻的倾倒分布。因此，罕见的事件形状的缩放模式的大部分，并保留有效指数的意义，这可以确定的基础上的数值和精确的结果。
Universality Classes in Isotropic, Abelian and non-Abelian, Sandpile Models
在各向同性，阿贝尔和非阿贝尔，沙堆模型的普遍性进行了研究，使用广泛的数值模拟。为了表征临界行为，我们采用了一组扩展的临界指数，雪崩的几何特征，以及描述雪崩过程中的面积和大小等平均量的时间演化的标度函数。比较了交换Bak-Tang-Wiesenfeld模型[P.Bak，C. Tang和K. Wiensenfeld，Phys. Rev. Lett. 59，381（1987）]和Manna [S. S. Manna，J. Phys. A. 24，L363（1991）]和Zhang [Y. C. Zhang，Phys. Rev. Lett. 63，470（1989）]我们发现强有力的迹象表明，这些模型中的每一个属于一个独特的普遍性类。
Relevance of Abelian Symmetry and Stochasticity in Directed Sandpiles
我们提供了一个全面的观点阿贝尔对称性和随机性的作用，在普遍性类的定向沙堆模型，在亚稳态模式和疤痕的潜在空间相关性的背景下。有人认为，阿贝尔对称性的相关性可能取决于是否动态规则是随机的或确定性的，通过亚稳模式和雪崩流的相互作用。基于新的标度关系，我们推测临界指数的雪崩，这是相当好地证实了在大规模的数值模拟。
Expansion and Contraction of Avalanches in 2D Abelian Sandpile
我们提出了一个详细的分析大规模模拟雪崩的2D阿贝尔沙堆模型。我们比较两种不同的分解成集群的倾倒和波的倾倒雪崩的统计特性。引入了辅助临界指数，并证实了Priezzhev等在PRL76，2093（1996）中提出的控制雪崩收缩的指数的存在。我们也给出了更详细的论证的确切值的指数表征波的统计。
Dissipative Abelian Sandpiles and Random Walks
我们证明了图L上的耗散阿贝尔沙堆可以与图L上的随机游动有关，图L由一个捕获点扩展而成。从这个关系式可以看出，在标度假设下，耗散沙堆的关联长度指数ν总是等于1/d_w，其中d_w是随机游动的分形维数。这导致了对已知结果的新的理解，即在任何欧几里得格上ν=1/2。然而，我们的结果是更普遍的，作为一个例子，我们也提出了确切的数据有限Sierpinski垫片，充分证实了我们的预测。
The abelian sandpile and related models
阿贝尔沙堆模型是自组织临界性的最简单易分析模型。本文简要回顾了有关该模型的已知结果。阿贝尔群结构允许精确计算它的许多性质。特别地，可以计算所有维度中有向模型的所有临界指数。对于无向情形，该模型与q= 0的Potts模型有关.这使得在二维中的一些指数的精确计算，并有一些关于其他的图解。我们还讨论了一个网络的通信反应式处理器的模型的推广。这包括沙堆模型与随机倾倒规则作为一个特殊的情况。我们还考虑了一个非阿贝尔随机变量，这是在一个不同的普遍性类，有关定向渗流。
Large scale simulations of the Zhang sandpile model
本文考虑Y.- C.二维正方形格子上的张。研究了该模型的静态和动态特性，并与Bak、Tang和Wiesenfeld的阿贝尔沙堆模型进行了比较。考虑到有限尺寸效应的详细分析表明，雪崩概率分布的指数与阿贝尔模型中的指数相同。
The Upper Critical Dimension of the Abelian Sandpile Model
现有的阿贝尔沙堆模型的上临界尺寸的估计是基于雪崩的定性考虑，作为自我避免分支过程。我们发现一个确切的表示雪崩作为一个序列的两个组件的生成树的生成子树。利用生成树上的化学路径与环擦除随机游动之间的等价性，我们将问题归结为生成子树的分形维数的确定。然后，上临界维数$d_u=4$遵循随机游动和环擦除随机游动的相交概率的Lawler定理。
Residence Time Distribution of Sand Grains in the 1-Dimensional Abelian Sandpile Model
本文研究了一维阿贝尔沙堆模型中沙粒在$L$格点上的停留时间$T$的概率分布，其中包括$T<<L^2$和$T>>L^2$ .分布函数衰减为$\exp(-\frac{K_LT}{L^2})$。我们数值计算系数$K_L$的值为$L$高达150。有趣的是，对于大的$L$，分布函数具有缩放形式$\frac{1}{L^a}f(\frac{T}{L^b})$和$a \neq b$。
We study the directed Abelian sandpile model on a square lattice, with $K$ downward neighbors per site, $K > 2$. The $K=3$ case is solved exactly, which extends the earlier known solution for the $K=2$ case. For $K>2$, the avalanche clusters can have holes and side-branches and are thus qualitatively different from the $K=2$ case where avalance clusters are compact. However, we find that the critical exponents for $K>2$ are identical with those for the $K=2$ case, and the large scale structure of the avalanches for $K>2$ tends to the $K=2$ case.
Time inhomogeneous Fokker-Plank equation for wave distributions in the Abelian sandpile model
阿贝尔沙堆模型中倾倒波的时间和尺度分布表示为尺度不变、时间非齐次的Fokker-Planck方程的初至原点分布。假设一个线性猜想的时间不均匀性指数的函数循环擦除随机游走（LERW）的临界指数，建议的数值结果，这种方法允许估计的下临界维数的模型和精确值的临界指数的LERW在三维。在二维中的雪崩尺寸分布被发现是两个封闭的幂律之间的差异。
A popular theory of self-organized criticality relates driven dissipative systems to systems with conservation. This theory predicts that the stationary density of the abelian sandpile model equals the threshold density of the fixed-energy sandpile. We refute this prediction for a wide variety of underlying graphs, including the square grid. Driven dissipative sandpiles continue to evolve even after reaching criticality. This result casts doubt on the validity of using fixed-energy sandpiles to explore the critical behavior of the abelian sandpile model at stationarity.
We introduce a natural stochastic extension, called SSP, of the abelian sandpile model(ASM), which shares many mathematical properties with ASM, yet radically differs in its physical behavior, for example in terms of the shape of the steady state and of the avalanche size distribution. We establish a basic theory of SSP analogous to that of ASM, and present a brief numerical study of its behavior. Our original motivation for studying SSP stems from its connection to the LLL algorithm established in another work by the authors [5]. The importance of understanding how LLL works cannot be stresse…
Universality classes in directed sandpile models
我们进行大规模的数值模拟的两个状态的随机沙堆模型的有向版本。数值结果表明，该随机模型定义了一个新的关于阿贝尔有向沙堆的普适类。不同的临界行为的物理起源归因于随机模型中的多个倾倒的存在。这些结果提供了新的见解长期争论的问题，在阿贝尔和随机沙堆的普遍性。
The aim of this note is to systematize our knowledge about identical configurations of ASM.
Time transport correlations in abelian sandpile models
本文研究了阿贝尔沙堆模型中时间输运相关性的通用性质，揭示了其与稳态性质的联系，并通过线性方程组建立了两点时间输运相关性与二阶矩之间的关系。
Steady state of Stochastic Sandpile Models
研究了具有随机倾倒规则的阿贝尔沙堆模型的稳定性。粒子加法算子彼此交换，但一般来说这些算子不需要是可对角化的。我们使用它们的交换代数来确定它们的特征值，以及Jordan块结构。然后，这些被用来确定在稳定状态下的不同配置的概率。我们说明这个过程中明确确定的数值精确的稳定状态的一维例子，系统的大小$\le12$，并研究在稳定状态下的密度分布。
Renormalization group approach to an Abelian sandpile model on planar lattices
重整化群（RG）方法求解格子沙堆模型的一个重要步骤是精确计算RG变换单元内沙堆动力学的所有可能的倾倒过程。在这里，我们提出了一个计算机算法来执行这样的精确计数的平面晶格的细胞在RG方法的Bak-Tang-Wiesenfeld沙堆模型[物理。{ \bf 59}，381（1987）]，并考虑由Pietronero，Vespignani和Zaplani（PVZ）提出的简化高RG方程[Phys. Rev. Lett. { \bf 72}，1690（1994）]和Ivashkevich [Phys. Rev. Lett. { \bf 76}，3368（1996）]。使用这种算法，我们能够更快地进行RG变换与大的细胞大小，例如$3 \times 3$细胞的正方形（sq）格在PVZ RG方程，这是目前最大的细胞大小，并发现一些错误，在以前的文件[物理评论E { \bf 51}，1711（1995）]。对于正方形格和平面三角形格，我们得到了每个格的唯一吸引不动点，并计算了雪崩指数$τ$和动力学指数$z$。我们的研究结果表明，在PVZ RG转化中细胞大小的增加不会导致更准确的结果。讨论了这一结果的意义。
The Abelian sandpile process evolves configurations of chips on the integer lattice by toppling any vertex with at least 4 chips, distributing one of its chips to each of its 4 neighbors. When begun from a large stack of chips, the terminal state of the sandpile has a curious fractal structure which has remained unexplained. Using a characterization of the quadratic growths attainable by integer-superharmonic functions, we prove that the sandpile PDE recently shown to characterize the scaling limit of the sandpile admits certain fractal solutions, giving a precise mathematical perspective on t…
Scaling fields in the two-dimensional abelian sandpile model
从二维共形场论的角度考虑了各向同性的二维阿贝尔沙堆模型。我们计算晶格关联函数的各种集群变量（在和关闭的临界），从中我们推断的领域的理论描述的标度限制。我们发现一个完美的协议与预测的c=-2共形场论和它的大规模扰动，从而提供了直接的证据，共形不变性和更普遍的描述方面的一个本地场论。高度2变量的问题也得到了解决，但还没有明确的结论。
Avalanche frontiers in Abelian Sandpile Model (ASM) are random simple curves whose continuum limit is known to be a Schramm-Loewner Evolution (SLE) with diffusivity parameter $κ= 2$. In this paper we consider the dissipative ASM and study the statistics of the avalanche and wave frontiers for various rates of dissipation. We examine the scaling behavior of a number of functions such as the correlation length, the exponent of distribution function of loop lengths and gyration radius defined for waves and avalanches. We find that they do scale with the rate of dissipation. Two significant length…
Boundary height fields in the Abelian sandpile model
本文研究了上半平面上的阿贝尔沙堆模型，重新考虑了边界上四个高度变量的相关性。为了方便起见，我们在模型的耗散（大规模）扩展中进行了分析，并确定了对应于四个高度的边界标度场。我们发现它们都可以用c=-2对数共形场论的大扰动来解释。
Universality Class of One-Dimensional Directed Sandpile Models
本文介绍了一个一般的n态有向沙堆模型。对于n < infty，导出了n态模型的平稳性质，基于中心极限定理的分析论证表明该模型属于完全非对称Oslo模型的普适类，对于小系统尺寸，该模型具有向不相关分支过程行为的交叉.因此，中心极限定理使我们能够确定一个大的普遍性类的一维有向沙堆模型的存在。
Scaling of Extreme Events in 2d BTW Sandpile
本研究针对二维Abelian沙堆模型的极端雪崩事件，采用块极大值法揭示其重标度分布分别收敛于Gumbel和Weibull分布族，提出标度函数并估算标度指数，为复杂系统极端事件统计研究提供框架。
We study universality classes and crossover behaviors in non-Abelian directed sandpile models, in terms of the metastable pattern analysis. The non-Abelian property induces spatially correlated metastable patterns, characterized by the algebraic decay of the grain density along the propagation direction of an avalanche. Crossover scaling behaviors are observed in the grain density due to the interplay between the toppling randomness and the parity of the threshold value. In the presence of such crossovers, we show that the broadness of the grain distribution plays a crucial role in resolving t…
We show that a fixed set of woven defect lines in a nematic liquid crystal supports a set of non-singular topological states which can be mapped on to recurrent stable configurations in the Abelian sandpile model or chip-firing game. The physical correspondence between local Skyrmion flux and sandpile height is made between the two models. Using a toy model of the elastic energy, we examine the structure of energy minima as a function of topological class and show that the system admits domain wall Skyrmion solitons.
The abelian sandpile model defines a Markov chain whose states are integer-valued functions on the vertices of a simple connected graph $G$. By viewing this chain as a (nonreversible) random walk on an abelian group, we give a formula for its eigenvalues and eigenvectors in terms of `multiplicative harmonic functions' on the vertices of $G$. We show that the spectral gap of the sandpile chain is within a constant factor of the length of the shortest non-integer vector in the dual Laplacian lattice, while the mixing time is at most a constant times the smoothing parameter of the Laplacian latti…
We check the universality properties of the two-dimensional Abelian sandpile model by computing some of its properties on the honeycomb lattice. Exact expressions for unit height correlation functions in presence of boundaries and for different boundary conditions are derived. Also, we study the statistics of the boundaries of avalanche waves by using the theory of SLE and suggest that these curves are conformally invariant and described by SLE2.
The height probabilities for the recurrent configurations in the Abelian Sandpile Model on the square lattice have analytic expressions, in terms of multidimensional quadratures. At first, these quantities have been evaluated numerically with high accuracy, and conjectured to be certain cubic rational-coefficient polynomials in 1/pi. Later their values have been determined by different methods. We revert to the direct derivation of these probabilities, by computing analytically the corresponding integrals. Yet another time, we confirm the predictions on the probabilities, and thus, as a coroll…
Fine Structure of Avalanches in the Abelian Sandpile Model
本文研究了线性尺寸为L的正方形格子上的二维阿贝尔沙堆模型。我们介绍了雪崩的精细结构的概念，并比较了雪崩和倾倒波的行为。我们表明，根据雪崩的精细结构的复杂程度，这是连续波的边界的复杂叠加的直接后果，雪崩分为两个不同的类别。我们建议缩放ansätz这些雪崩类型，并验证它们的数值。我们发现，虽然第一种类型的雪崩有一个简单的标度行为，第二（复杂）类型的特点是由雪崩大小相关的标度指数。这提供了一个框架，在该框架内，人们可以理解该模型中一致的缩放行为的失败。
On the scaling behavior of the abelian sandpile model
二维阿贝尔沙堆模型不具有平衡系统的临界行为。更确切地说，稳态的性质来自于这样一个条件，即在距离系统边界r处开始的雪崩到达边界的概率与1/sqrt（r）成正比。因此，该模型的标度行为可以从单独评估耗散雪崩获得，不仅允许确定所有指数的值，但也显示了有限尺寸标度的故障。
The Abelian Sandpile Model is a cellular automaton whose discrete dynamics reaches an out-of-equilibrium steady state resembling avalanches in piles of sand. The fundamental moves defining the dynamics are encoded by the toppling rules. The transition monoid corresponding to this dynamics in the set of stable configurations is abelian, a property which seems at the basis of our understanding of the model. By including also antitoppling rules, we introduce and investigate a larger monoid, which is not abelian anymore. We prove a number of algebraic properties of this monoid, and describe their …
Logarithmic scaling for height variables in the Abelian sandpile model
我们报告的1点函数的标度形式的精确计算，在上半平面上，高度2变量的二维阿贝尔沙堆模型。通过比较开边界条件和闭边界条件，我们发现与高度2相关的标度场是标度维数为2的对数标量场，属于c=-2对数共形场论。这种识别是由数值模拟证实，并扩展到高度3和4变量，表现出相同的缩放形式。使用保角设置，我们提出了精确的建议，所有高度变量的批量2点函数。
Explicit characterization of the identity configuration in an Abelian Sandpile Model
自从Creutz的工作以来，确定阿贝尔沙堆模型（ASM）在给定格点上的群恒等式是一个令人困惑的问题：在Z^2的矩形部分上，出现了复杂的准自相似结构。我们研究的ASM上的正方形格，在不同的几何形状，并与定向边的变体。通过它们的额外对称性，可以很容易地确定身份，这是一个齐次函数。正方形几何上的有向变体显示出显著的精确结构，渐近自相似。
One-Dimensional Directed Sandpile Models and the Area under a Brownian Curve
我们导出了一维一般有向“沙堆”模型的定态性质。使用中心极限定理的依赖随机变量，我们找到了精确的条件，该模型属于普遍性类的完全不对称奥斯陆模型，从而确定了一个大的普遍性类的有向沙堆。我们将雪崩大小映射到布朗曲线下的区域，并在原点处具有吸收边界，从而激励我们解决这个布朗曲线问题。因此，我们能够确定在这个普适性类的雪崩大小概率的矩生成函数，明确计算的领先阶项的振幅。
Abelian Sandpiles on Cylinders
本文研究圆柱面上宽度远大于周长时的阿贝尔沙堆模型，发现雪崩规模分布呈现阶梯结构，并解释了首级阶梯内雪崩基本等概率及其小指数尾部现象。
We review the status of the two-dimensional Abelian sandpile model as a strong candidate to provide a lattice realization of logarithmic conformal invariance with central charge c=-2. Evidence supporting this view is collected from various aspects of the model. These include the study of some conformally invariant boundary conditions, and the corresponding boundary condition changing fields, the calculation of correlations of certain bulk and boundary observables (the height variables) as well as a proper account of the necessary dissipation, which allows for a physical understanding of some o…
The Abelian Sandpile generates complex and beautiful patterns and seems to display allometry. On the plane, beyond patches, patterns periodic in both dimensions, we remark the presence of structures periodic in one dimension, that we call strings. We classify completely their constituents in terms of their principal periodic vector k, that we call momentum. We derive a simple relation between the momentum of a string and its density of particles, E, which is reminiscent of a dispersion relation, E=k^2. Strings interact: they can merge and split and within these processes momentum is conserved.…
Continuously varying exponents in a sandpile model with dissipation near surface
本文考虑了有向阿贝尔沙堆模型中存在汇点的情形，汇点的密度f_t在上表面以下的深度t处随c~1/t^chi变化。对于chi>1，无序是不相关的。对于chi<1，它是相关的，并且模型对于任何非零的c不再是关键的。对于chi=1，雪崩分布的指数连续地依赖于无序的振幅c。我们精确地计算了这种依赖关系，并通过模拟验证了结果。
Wind on the boundary for the Abelian sandpile model
我们继续我们的研究的二维阿贝尔沙堆模型的对数共形场论与中心荷c=-2，通过引入两个新的边界条件。它们有两个不寻常的特征：它们带有内在的方向，更奇怪的是，它们不能均匀地施加在整个边界上（如圆柱体的边缘）。它们导致七个新的边界条件变化的字段，其中一些是在最高权表示（重量-1/8，0和3/8），其他一些属于不可分解的表示秩2 Jordan细胞（最低的重量0和1）。他们的融合代数似乎完全符合Gaberdiel和Kausch提出的融合规则。
Avalanches and Waves in the Abelian Sandpile Model
本文用数值方法研究了二维阿贝尔沙堆模型中的崩塌问题。Priezzhev等人[PRL 76，2093（1996）]最近提出了该模型中临界指数的精确结果，该模型基于后续波大小差异的建议标度关系$Δs =s_{k}- s_{k+1}$的存在，其中前一波$s_{k}$的大小被认为几乎总是下一波$s_{k+1}$大小的上限。在这里，我们表明，$Δs$的显着贡献来自波，违反了限制，平均$<Δs(s_{k})>$实际上是负的，发散与系统的大小，矛盾的解决方案。
A popular theory of self-organized criticality relates the critical behavior of driven dissipative systems to that of systems with conservation. In particular, this theory predicts that the stationary density of the abelian sandpile model should be equal to the threshold density of the corresponding fixed-energy sandpile. This "density conjecture" has been proved for the underlying graph Z. We show (by simulation or by proof) that the density conjecture is false when the underlying graph is any of Z^2, the complete graph K_n, the Cayley tree, the ladder graph, the bracelet graph, or the flower…
Pattern formation in growing sandpiles
在阿贝尔沙堆模型中，在平坦基底上的单个位置添加颗粒会产生美丽的复杂图案。我们详细研究了在二维正方形晶格上添加晶粒所产生的图案，该晶格具有定向边缘（每个站点有两个向内的箭头和两个向外的箭头），从占据一半站点的周期性背景开始。所形成的图案的尺寸与添加的颗粒的数量成比例，如$N$。我们给出精确的渐近模式的特征，在位置和形状的模式的不同功能。
The Abelian Sandpile Model, seen as a deterministic lattice automaton, on two-dimensional periodic graphs generates complex regular patterns displaying (fractal) self-similarity. In particular, on a variety of lattices and initial conditions, at all sizes, there appears what we call an exact Sierpinski structure: the volume is filled with periodic patterns, glued together along straight lines, with the topology of a triangular Sierpinski gasket. Various lattices (square, hexagonal, kagome,...), initial conditions, and toppling rules show Sierpinski structures which are apparently unrelated and…
Sandpile Model with Activity Inhibition
本文研究了一种新的沙堆模型，在该模型中，系统的键在经过一定数量的颗粒传输后被抑制活动。这种情况促使一个不稳定的沙柱只把颗粒分布到那些倾倒次数少于m次的相邻沙柱上。在这种非阿贝尔模型中，颗粒的有效移动速度比普通扩散（超扩散）快。一个新的系统大小依赖的交叉从阿贝尔沙堆行为的一个新的临界行为观察到的参数m的所有值。
We present the detailed calculations of the asymptotics of two-site correlation functions for height variables in the two-dimensional Abelian sandpile model. By using combinatorial methods for the enumeration of spanning trees, we extend the well-known result for the correlation $σ_{1,1} \simeq 1/r^4$ of minimal heights $h_1=h_2=1$ to $σ_{1,h} = P_{1,h}-P_1P_h$ for height values $h=2,3,4$. These results confirm the dominant logarithmic behaviour $σ_{1,h} \simeq (c_h\log r + d_h)/r^4 + {\cal O}(r^{-5})$ for large $r$, predicted by logarithmic conformal field theory based on field identification…
We consider patterns generated by adding large number of sand grains at a single site in an abelian sandpile model with a periodic initial configuration, and relaxing. The patterns show proportionate growth. We study the robustness of these patterns against different types of noise, \textit{viz.}, randomness in the point of addition, disorder in the initial periodic configuration, and disorder in the connectivity of the underlying lattice. We find that the patterns show a varying degree of robustness to addition of a small amount of noise in each case. However, introducing stochasticity in the…
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…
The leaky abelian sandpile model (Leaky-ASM) is a growth model in which $n$ grains of sand start at the origin in $\mathbb{Z}^2$ and diffuse along the vertices according to a toppling rule. A site can topple if its amount of sand is above a threshold. In each topple a site sends some sand to each neighbor and leaks a portion $1-1/d$ of its sand. We compute the limit shape as a function of $d$ in the symmetric case where each topple sends an equal amount of sand to each neighbor. The limit shape converges to a circle as $d\to 1$ and a diamond as $d\to\infty$. We compute the limit shape by compa…
Universal correlations in the Abelian sandpile model
本文数值研究二维阿贝尔沙堆模型的体关联函数，验证对数共形场论预测，并首次对kagome格进行系统数值分析，同时推导了高度-1概率的闭式解析表达式。
We consider the Abelian sandpile model on triangular and hexagonal lattices. We compute several height probabilities on the full plane and on half-planes, and discuss some properties of the universality of the model.
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