On the Ornstein-Zernike behaviour for the supercritical Random-Cluster model on $\mathbb{Z}^{d},d\geq3.$

On the Ornstein-Zernike equation for stationary cluster processes and the random connection model

In the first part of this paper we consider a general stationary subcritical cluster model in $\mathbb{R}^d$. The associated pair-connectedness function can be defined in terms of two-point Palm probabilities of the underlying point process. Using Palm calculus and Fourier theory we solve the Ornstein-Zernike equation (OZE) under quite general distributional assumptions. In the second part of the paper we discuss the analytic and combinatorial properties of the OZE-solution in the special case of a Poisson driven random connection model.

具有无穷程相互作用的Ising模型的Ornstein-Zernike行为

Ornstein-Zernike behavior for Ising models with infinite-range interactions

本文首次在无穷程相互作用下证明了Ising模型两点函数的Ornstein-Zernike渐近行为，并将结果推广至包含self-avoiding walk等在内的平均自排斥模型族。

从亚临界到临界衰减的交叉：随机游走、自避游走、渗透

Crossover from subcritical to critical decay: random walk, self-avoiding walk, percolation

本文研究了平衡统计力学中亚临界两点函数的奥恩斯坦-泽恩基衰减，证明了在适当假设下， Ornstein-Zernike方程在Z^d上的解行为与布朗运动带漂移的格林函数乘以各向异性指数衰减因子一致，适用于多种随机游走、自避游走和渗透模型。

Spatial CLT for the supercritical Ornstein-Uhlenbeck superprocess

In this paper we consider a superprocess being a measure-valued diffusion corresponding to the equation $u_{t}=Lu+αu-βu^{2}$, where $L$ is the infinitesimal operator of the \emph{Ornstein-Uhlenbeck process} and $β>0,\:α>0$. The latter condition implies that the process is \emph{supercritical,} i.e. its total mass grows exponentially. This system is known to fulfill a law of large numbers. In the paper we prove the corresponding \emph{central limit theorem}. The limit and the CLT normalization fall into three qualitatively different classes. In what we call the small growth rate case the situat…

超临界 Bernoulli 逾渗在 $\mathbb{Z}^d$（$d\geq 3$）上的非微扰有限连通性

Non-perturbative finite connectivities in supercritical Bernoulli percolation on $\mathbb{Z}^d$ with $d\geq 3$

本文解决了超临界 Bernoulli 逾渗中有限连通概率的长期问题，通过更新结构与改进的粗粒化方法，推导出尖锐的 Ornstein--Zernike 渐近形式。

Density fluctuations for a zero-range process on the percolation cluster

We prove that the density fluctuations for a zero-range process evolving on the supercritical percolation cluster are given by a generalized Ornstein-Uhlenbeck process in the space of distributions $\mc S'(\bb R^d)$.

The Direct-Connectedness Function in the Random Connection Model

We investigate expansions for connectedness functions in the random connection model of continuum percolation in powers of the intensity. Precisely, we study the pair-connectedness and the direct-connectedness functions, related to each other via the Ornstein-Zernike equation. We exhibit the fact that the coefficients of the expansions consist of sums over connected and $2$-connected graphs. In the physics literature, this is known to be the case more generally for percolation models based on Gibbs point processes and stands in analogy to the formalism developed for correlation functions in li…

Fluctuation theory of connectivities for subcritical random cluster models

We develop a fluctuation theory of connectivities for subcritical random cluster models. The theory is based on a comprehensive nonperturbative probabilistic description of long connected clusters in terms of essentially one-dimensional chains of irreducible objects. Statistics of local observables, for example, displacement, over such chains obey classical limit laws, and our construction leads to an effective random walk representation of percolation clusters. The results include a derivation of a sharp Ornstein--Zernike type asymptotic formula for two point functions, a proof of analyticity…