Erik I. Broman, Jeffrey E. Steif
In this paper we will investigate dynamic stability of percolation for the stochastic Ising model and the contact process. We also introduce the notion of downward and upward $ε$-movability which will be a key tool for our analysis.
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In a recent paper by two of the authors, the concepts of upwards and downwards $ε$-movability were introduced, mainly as a technical tool for studying dynamical percolation of interacting particle systems. In this paper, we further explore these concepts which can be seen as refinements or quantifications of stochastic domination, and we relate them to previously studied concepts such as uniform insertion tolerance and extractability.
The aim of this paper is to underline the relation between reversible growth processes and invariant percolation. We present two models of interacting branching random walks (BRWs), truncated BRWs and competing BRWs, where survival of the growth process can be formulated as the existence of an infinite cluster in an invariant percolation on a tree. Our approach is fairly conceptual and allows generalizations to a wider set of "reversible" growth processes.
Interacting particle systems and percolation have been among the most active areas of probability theory over the past half century. Ted Harris played an important role in the early development of both fields. This paper is a bird's eye view of his work in these fields, and of its impact on later research in probability theory and mathematical physics.
Percolation representations of additive particle systems
本文探讨了有限分配格局部状态空间下加性相互作用粒子系统的一般渗透表示构造方法，并以Krone两阶段接触过程为例进行阐述。
We discuss the interrelation between phase transitions in interacting lattice or continuum models, and the existence of infinite clusters in suitable random-graph models. In particular, we describe a random-geometric approach to the phase transition in the continuum Ising model of two species of particles with soft or hard interspecies repulsion. We comment also on the related area-interaction process and on perfect simulation.
We consider an interacting particle system on the lattice involving pushing and blocking interactions, called PushASEP, in the presence of a wall at the origin. We show that the invariant measure of this system is equal in distribution to a vector of point-to-line last passage percolation times in a random geometrically distributed environment. The largest co-ordinates in both of these vectors are equal in distribution to the all-time supremum of a non-colliding random walk.
We study an interacting particle system in which moving particles activate dormant particles linked by the components of critical bond percolation. Addressing a conjecture from Beckman, Dinan, Durrett, Huo, and Junge for a continuous variant, we prove that the process can reach infinity in finite time i.e., explode. In particular, we prove that explosions occur almost surely on regular trees as well as oriented and unoriented two-dimensional integer lattices with sufficiently many particles per site. The oriented case requires an additional hypothesis about the existence and value of a certain…
Quantitative propagation of chaos for non-exchangeable diffusions via first-passage percolation
本文发展了非渐近方法，用于分析n个相互作用的扩散粒子系统，研究非交换性相互作用对边际分布与产品测度的相对熵估计。
Existence and Convergence of Interacting Particle Systems on Graphs
本文研究了图上相互作用粒子系统在无界度数或跳跃速率下的存在性和收敛性，通过引入新的假设和条件，证明了在某些随机图模型中这些系统能够存在。
A Course in Interacting Particle Systems
本讲义系统介绍相互作用粒子系统的核心理论工具，并通过数值模拟展示经典模型之外的丰富例子。
Uniqueness of the infinite cluster for monotone percolation models without insertion tolerance
本文研究了一类依赖性位点渗流模型，证明在特定条件下超临界相中存在唯一的无限簇，应用于阿贝尔沙堆等模型。
On the duality between particles and polymers
本文研究了粒子系统与聚合物模型之间的对偶性关系，通过分析三种模型得出了相关确定性边界的行列式公式。
We describe, at the microscopic level, the dynamics of N interacting components where the probability is very small when N is large that a given component interact more than once, directly or indirectly, up to time t, with any other component. Due to this fact, we can consider, at the macroscopic level, the quadratic system of differential equations associated with the interaction and establish an explicit formula for the solution of this system. We moreover apply our results to some models of Econophysics.
A quantization of interacting particle systems
本文将包括Domany-Kinzel模型的相互作用粒子系统视为图上的马尔可夫链，提出一种新的量子化方法，并引入与计算了量子化模型的zeta函数及绝对zeta函数。
Kinetically constrained models (KCM) are reversible interacting particle systems on $\mathbb Z^d$ with continuous time Markov dynamics of Glauber type, which represent a natural stochastic (and non-monotone) counterpart of the family of cellular automata known as $\mathcal U$-bootstrap percolation. KCM also display some of the peculiar features of the so-called "glassy dynamics", and as such they are extensively used in the physics literature to model the liquid-glass transition, a major and longstanding open problem in condensed matter physics. We consider two-dimensional KCM with update rule…
In this paper we will show how the results found in Cator and Pimentel 2009, about the Busemann functions in last-passage percolation, can be used to calculate the asymptotic distribution of the speed of a single second class particle starting from an arbitrary deterministic configuration which has a rarefaction fan, in either the totally asymetric exclusion process, or the Hammersley interacting particle process. The method will be to use the well known last-passage percolation description of the exclusion process and of the Hammersley process, and then the well known connection between secon…
Hydrodynamic Limits of non-Markovian Interacting Particle Systems on Sparse Graphs
本文研究了稀疏图上非马尔可夫交互粒子系统的流体动力学极限，证明了在温和条件下经验测度弱收敛到根顶点边缘动力学，并引入了有限可分解性等概念确保系统适定性与渐近相关性衰减。
Bi-infinite systems of singularly interacting Brownian particles and the KPZ equation
研究实线上具奇异非对称相互作用的布朗粒子双无穷系统，通过碰撞链局部化等方法，建立适定性理论与确定平稳分布，证明在弱非对称下粒子数函数平衡涨落收敛到乘性随机热方程，将其置于KPZ普适类。
Exact solution of interacting particle systems related to random matrices
本文研究具有多项式漂移和扩散系数的一维扩散粒子系统，利用显式核的Fredholm行列式求得其有限维分布，并证明非碰撞扩散模型具有行列式相关函数，为空间非均匀条件下的一般初始条件提供了精确解。
We present a perfect simulation algorithm for measures that are absolutely continuous with respect to some Poisson process and can be obtained as invariant measures of birth-and-death processes. Examples include area- and perimeter-interacting point processes (with stochastic grains), invariant measures of loss networks, and the Ising contour and random cluster models. The algorithm does not involve couplings of the process with different initial conditions and it is not tied up to monotonicity requirements. Furthermore, it directly provides perfect samples of finite windows of the infinite-vo…
The critical length for growing a droplet
本文研究相互作用粒子系统中液滴出现的临界长度，为 d 维格点上任意有限范围单调元胞自动机构建液滴生长机制，并得到渗流临界概率的近精确上界。
Kinetically constrained models (KCM) are reversible interacting particle systems on $\mathbb{Z}^d$ with continuous-time constrained Glauber dynamics. They are a natural non-monotone stochastic version of the family of cellular automata with random initial state known as $\mathcal{U}$-bootstrap percolation. KCM have an interest in their own right, owing to their use for modelling the liquid-glass transition in condensed matter physics. In two dimensions there are three classes of models with qualitatively different scaling of the infection time of the origin as the density of infected sites van…
The extreme statistics of some noncolliding Brownian processes
研究非碰撞粒子系统极限理论，揭示布朗运动最大特征值的极限行为、Airy过程极限及非碰撞布朗桥最大路径的Fredholm行列式公式。
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