Christian Maes, Frank Redig, Ellen Saada
We consider the standard Abelian sandpile process on the Bethe lattice. We show the existence of the thermodynamic limit for the finite volume stationary measures and the existence of a unique infinite volume Markov process exhibiting features of self-organized criticality.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Random walks in a field of soft traps and criticality for the dissipative Abelian Sandpile Model
该研究受耗散阿贝尔沙堆模型启发，分析一维随机游走在间距递增软陷阱景观中的轨迹，确定了连续陷阱间区间大小的临界增长率，揭示其对应沙堆模型的临界性转变，精确量化了破坏原始模型临界性所需的耗散量。
In their previous work, the authors studied the abelian sandpile model on graphs constructed from a growing piece of a plane or space tiling, given periodic or open boundary conditions, and identified spectral factors which govern the asymptotic spectral gap and asymptotic mixing time. This article gives a general method of determining the spectral factor either computationally or asymptotically and performs the determination in specific examples.
We prove that the Abelian sandpile model on a random binary and binomial tree, as introduced in \cite{rrs}, is not critical for all branching probabilities $p<1$; by estimating the tail of the annealed survival time of a random walk on the binary tree with randomly placed traps, we obtain some more information about the exponential tail of the avalanche radius. Next we study the sandpile model on $\mathbb{Z}^d$ with some additional dissipative sites: we provide examples and sufficient conditions for non-criticality; we also make a connection with the parabolic Anderson model. Finally we initia…
We analyse the abelian sandpile model on $\mathbbm{Z}^d$ for the starting configuration of $n$ particles in the origin and $2d-2$ particles otherwise. We give a new short proof of the theorem of Fey, Levine and Peres \cite{FLP} that the radius of the toppled cluster of this configuration is $O(n^{1/d})$.
We define stabilizability of an infinite volume height configuration and of a probability measure on height configurations. We show that for high enough densities, a probability measure cannot be stabilized. We also show that in some sense the thermodynamic limit of the uniform measures on the recurrent configurations of the abelian sandpile model (ASM) is a maximal element of the set of stabilizable measures. In that sense the self-organized critical behavior of the ASM can be understood in terms of an ordinary transition between stabilizable and non-stabilizable
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…
We present a construction of an entropy-preserving equivariant surjective map from the $d$-dimensional critical sandpile model to a certain closed, shift-invariant subgroup of $\mathbb{T}^{\mathbb{Z}^d}$ (the `harmonic model'). A similar map is constructed for the dissipative abelian sandpile model and is used to prove uniqueness and the Bernoulli property of the measure of maximal entropy for that model.
We study the Abelian sandpile model on Z^d. In dimensions at least 3 we prove existence of the infinite volume addition operator, almost surely with respect to the infinite volume limit mu of the uniform measures on recurrent configurations. We prove the existence of a Markov process with stationary measure mu, and study ergodic properties of this process. The main techniques we use are a connection between the statistics of waves and uniform two-component spanning trees and results on the uniform spanning tree measure on Z^d.
We study the abelian sandpile model on a random binary tree. Using a transfer matrix approach introduced by Dhar & Majumdar, we prove exponential decay of correlations, and in a small supercritical region (i.e., where the branching process survives with positive probability) exponential decay of avalanche sizes. This shows a phase transition phenomenon between exponential decay and power law decay of avalanche sizes. Our main technical tools are: (1) A recursion for the ratio between the numbers of weakly and strongly allowed configurations which is proved to have a well-defined stochastic sol…
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。
Fermionic Gaussian free field structure in the Abelian sandpile model and uniform spanning tree
本文用费米子高斯自由场严格构造了Abelian沙堆模型高度为1的场与一致生成树度场的有限体积表示，据此计算了累积量并证明了其在正方形与三角格点上的标度极限普适性。
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.
Optimal local interventions in the two-dimensional Abelian sandpile model
研究通过局部干预减少沙堆模型中灾难性事件规模，分析干预策略对最大雪崩规模和缓解雪崩数量的影响。
We investigate the avalanche dynamics of the abelian sandpile model on arbitrarily large balls of the expanded cactus graph (the Cayley graph of the free product $\mathbb{Z}_3 * \mathbb{Z}_2$). We follow the approach of Dhar and Majumdar (1990) to enumerate the number of recurrent configurations. We also propose the filling method of enumerating all the recurrent configurations in which adding a grain to a designated origin vertex (far enough away from the boundary vertices) causes topplings to occur in a specific cluster (a connected subgraph that is the union of cells, or copies of the 3-cyc…
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.
Limit shape of the leaky Abelian sandpile model with multiple layers
本文在任意维度上推广渗漏阿贝尔沙堆模型，引入多层颜色与任意倾倒分布，证明极限形状存在，并展示其随参数变化收敛到多面体或椭球，且具有二元性。
This survey is an extended version of lectures given at the Cornell Probability Summer School 2013. The fundamental facts about the Abelian sandpile model on a finite graph and its connections to related models are presented. We discuss exactly computable results via Majumdar and Dhar's method. The main ideas of Priezzhev's computation of the height probabilities in 2D are also presented, including explicit error estimates involved in passing to the limit of the infinite lattice. We also discuss various questions arising on infinite graphs, such as convergence to a sandpile measure, and stabil…
Since the work of Creutz, identifying the group identities for the Abelian Sandpile Model (ASM) on a given lattice is a puzzling issue: on rectangular portions of Z^2 complex quasi-self-similar structures arise. We study the ASM on the square lattice, in different geometries, and a variant with directed edges. Cylinders, through their extra symmetry, allow an easy determination of the identity, which is a homogeneous function. The directed variant on square geometry shows a remarkable exact structure, asymptotically self-similar.
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…
We continue our investigation of the two-dimensional Abelian sandpile model in terms of a logarithmic conformal field theory with central charge c=-2, by introducing two new boundary conditions. These have two unusual features: they carry an intrinsic orientation, and, more strangely, they cannot be imposed uniformly on a whole boundary (like the edge of a cylinder). They lead to seven new boundary condition changing fields, some of them being in highest weight representations (weights -1/8, 0 and 3/8), some others belonging to indecomposable representations with rank 2 Jordan cells (lowest we…
Let Λbe a finite subset of Z^d. We study the following sandpile model on Λ. The height at any given vertex x of Λis a positive real number, and additions are uniformly distributed on some interval [a,b], which is a subset of [0,1]. The threshold value is 1; when the height at a given vertex exceeds 1, it topples, that is, its height is reduced by 1, and the heights of all its neighbours in Λincrease by 1/2d. We first establish that the uniform measure μon the so called "allowed configurations" is invariant under the dynamics. When a < b, we show with coupling ideas that starting from any initi…
We discuss various critical densities in sandpile models. The stationary density is the average expected height in the stationary state of a finite-volume model; the transition density is the critical point in the infinite-volume counterpart. These two critical densities were generally assumed to be equal, but this has turned out to be wrong for deterministic sandpile models. We show they are not equal in a quenched version of the Manna sandpile model either. In the literature, when the transition density is simulated, it is implicitly or explicitly assumed to be equal to either the so-called …
We introduce a family of abelian sandpile models with two parameters $n, m \in {\bf N}$ defined on finite lattices on $d$-dimensional torus. Sites with $2dn+m$ or more grains of sand are unstable and topple, and in each toppling $m$ grains dissipate from the system. Because of dissipation in bulk, the models are well-defined on the shift-invariant lattices and the infinite-volume limit of systems can be taken. From the determinantal expressions, we obtain the asymptotic forms of the avalanche propagators and the height-$(0,0)$ correlations of sandpiles for large distances in the infinite-volum…
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 study the sandpile model in infinite volume on $\mathbb{Z}^d$. In particular, we are interested in the question whether or not initial configurations, chosen according to a stationary measure $μ$, are $μ$-almost surely stabilizable. We prove that stabilizability does not depend on the particular procedure of stabilization we adopt. In $d=1$ and $μ$ a product measure with density $ρ=1$ (the known critical value for stabilizability in $d=1$) with a positive density of empty sites, we prove that $μ$ is not stabilizable. Furthermore, we study, for values of $ρ$ such that $μ$ is stabilizable, pe…
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…
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.
We compute the lattice 1-site probabilities, on the upper half-plane, of the four height variables in the two-dimensional Abelian sandpile model. We find their exact scaling form when the insertion point is far from the boundary, and when the boundary is either open or closed. Comparing with the predictions of a logarithmic conformal theory with central charge c=-2, we find a full compatibility with the following field assignments: the heights 2, 3 and 4 behave like (an unusual realization of) the logarithmic partner of a primary field with scaling dimension 2, the primary field itself being a…
We revisit the calculation of height correlations in the two-dimensional Abelian sandpile model by taking advantage of a technique developed recently by Kenyon and Wilson. The formalism requires to equip the usual graph Laplacian, ubiquitous in the context of cycle-rooted spanning forests, with a complex connection. In the case at hand, the connection is constant and localized along a semi-infinite defect line (zipper). In the appropriate limit of a trivial connection, it allows one to count spanning forests whose components contain prescribed sites, which are of direct relevance for height co…
We establish quantitative spherical shape theorems for rotor-router aggregation and abelian sandpile growth on the graphical Sierpinski gasket ($SG$) when particles are launched from the corner vertex. In particular, the abelian sandpile growth problem is exactly solved via a recursive construction of self-similar sandpile tiles. We show that sandpile growth and patterns exhibit a $(2\cdot 3^n)$-periodicity as a function of the initial mass. Moreover, the cluster explodes---increments by more than 1 in radius---at periodic intervals, a phenomenon not seen on $\mathbb{Z}^d$ or trees. We explici…
Operator formalism for discretely holomorphic parafermions of the two-color Ashkin-Teller, loop $\mathrm{O} ( 1 )$, staggered eight-vertex, odd eight-vertex, and abelian sandpile models
本研究将伊辛模型的算子形式体系推广至Ashkin-Teller、圈O(n)等模型，结合已有研究结论，建立了多类统计物理模型间的进一步关联，为相关严格分析提供基础。
We compute the correlations of two height variables in the two-dimensional Abelian sandpile model. We extend the known result for two minimal heights to the case when one of the heights is bigger than one. We find that the most dominant correlation log r/r^4 exactly fits the prediction obtained within the logarithmic conformal approach.
We define two general classes of nonabelian sandpile models on directed trees (or arborescences) as models of nonequilibrium statistical phenomena. These models have the property that sand grains can enter only through specified reservoirs, unlike the well-known abelian sandpile model. In the Trickle-down sandpile model, sand grains are allowed to move one at a time. For this model, we show that the stationary distribution is of product form. In the Landslide sandpile model, all the grains at a vertex topple at once, and here we prove formulas for all eigenvalues, their multiplicities, and the…
Single site height probabilities in the Abelian sandpile model, and the corresponding mean height $<h>$, are directly related to the probability $P_{\rm ret}$ that a loop erased random walk passes through a nearest neighbour of the starting site (return probability). The exact values of these quantities on the square lattice have been conjectured, in particular $<h> = 25/8$ and $P_{\rm ret} = 5/16$. We provide a rigourous proof of this conjecture by using a {\it local} monomer-dimer formulation of these questions.
The Abelian Sandpile Model is a discrete diffusion process defined on graphs (Dhar \cite{DD90}, Dhar et al. \cite{DD95}) which serves as the standard model of self-organized criticality. The transience class of a sandpile is defined as the maximum number of particles that can be added without making the system recurrent (\cite{BT05}). We demonstrate a class of sandpile which have polynomially bound transience classes by identifying key graph properties that play a role in the rapid diffusion process. These are the volume growth parameters, boundary regularity type properties and non-empty inte…
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。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。