On the thermodynamic limit for a one-dimensional sandpile process

可分沙堆的定量爆炸与渗流

Quantitative explosion and percolation of the divisible sandpile

针对Zd上的可分沙堆模型，在Levine等人证明均值为1时该过程会爆炸的基础上，本文确定了不同维度下某格点在时间t前释放质量的量级，还证明了对所有d≥2，该模型的 toppling 格点集在均值低于1时含无限连通分支，存在非平凡渗流相变，回答了相关问题。

Infinite volume limit of the Abelian sandpile model in dimensions d >= 3

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.

The abelian sandpile model on a random binary tree

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…

Convergence of the Abelian sandpile

The Abelian sandpile growth model is a diffusion process for configurations of chips placed on vertices of the integer lattice $\mathbb{Z}^d$, in which sites with at least 2d chips {\em topple}, distributing 1 chip to each of their neighbors in the lattice, until no more topplings are possible. From an initial configuration consisting of $n$ chips placed at a single vertex, the rescaled stable configuration seems to converge to a particular fractal pattern as $n\to \infty$. However, little has been proved about the appearance of the stable configurations. We use PDE techniques to prove that th…

Existence and uniqueness of the stationary measure in the continuous Abelian sandpile

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…

Functional Central Limit Theorems for Local Statistics of Spatial Birth-Death Processes in the Thermodynamic Regime

We present normal approximation results at the process level for local functionals defined on dynamic Poisson processes in $\mathbb{R}^d$. The dynamics we study here are those of a Markov birth-death process. We prove functional limit theorems in the so-called thermodynamic regime. Our results are applicable to several functionals of interest in the stochastic geometry literature, including subgraph and component counts in the random geometric graphs.

非竞争化学反应网络的阿贝尔结构

On the abelian structure of noncompetitive chemical reaction networks

本文研究非竞争化学反应网络的阿贝尔结构，证明其为阿贝尔网络的特殊实例，并建立统一的代数和概率分析框架。

Full Current Statistics for a Disordered Open Exclusion Process

We consider the nonabelian sandpile model defined on directed trees by Ayyer, Schilling, Steinberg and Thiéry (Commun. Math. Phys, 2013) and restrict it to the special case of a one-dimensional lattice of $n$ sites which has open boundaries and disordered hopping rates. We focus on the joint distribution of the integrated currents across each bond simultaneously, and calculate its cumulant generating function exactly. Surprisingly, the process conditioned on seeing specified currents across each bond turns out to be a renormalised version of the same process. We also remark on a duality proper…

Abelian oil and water dynamics does not have an absorbing-state phase transition

The oil and water model is an interacting particle system with two types of particles and a dynamics that conserves the number of particles, which belongs to the so-called class of Abelian networks. Widely studied processes in this class are sandpiles models and activated random walks, which are known (at least for some choice of the underlying graph) to undergo an absorbing-state phase transition. This phase transition characterizes the existence of two regimes, depending on the particle density: a regime of fixation at low densities, where the dynamics converges towards an absorbing state an…

Construction of the Thermodynamic Jamming Limit for the Parking Process and Other Exclusion Schemes on Z^d

We provide an explicit construction of the thermodynamic jamming limit for the parking process and other finite range exclusion schemes on Z^d. By means thereof, a strong law of large numbers for occupation densities is accomplished, and, amongst other results, the so called "super-exponential" (i.e. gamma) decay of pair correlation functions is established.

Non-fixation for conservative stochastic dynamics on the line

We consider Activated Random Walk (ARW), a model which generalizes the Stochastic Sandpile, one of the canonical examples of self organized criticality. Informally ARW is a particle system on $\mathbb{Z}$ with mass conservation. One starts with a mass density $μ>0$ of initially active particles, each of which performs a symmetric random walk at rate one and falls asleep at rate $λ>0$. Sleepy particles become active on coming in contact with other active particles. We investigate the question of fixation/non-fixation of the process, and show for small enough $λ$, the critical mass density for f…