Ornstein-Zernike Theory for the finite range Ising models above T_c

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.

平面随机团簇模型的近临界Ornstein-Zernike理论

Near-critical Ornstein--Zernike theory for the planar random-cluster model

针对1≤q<4的二维随机团簇模型，在近临界区域建立Ornstein-Zernike理论，证明两点函数在p<p_c时一致成立的渐近公式，通过相关长度尺度上的次临界渗流团簇更新性质分析，并导出逆相关长度严格凸性等结论。

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。

Sharp Asymptotics for the Truncated Two-Point Function of the Ising Model with a Positive Field

We prove that the correction to exponential decay of the truncated two points function in the homogeneous positive field Ising model is $c\|x\|^{-(d-1)/2}$. The proof is based on the development in the random current representation of a "modern" Ornstein-Zernike theory, as developed by Campanino, Ioffe and Velenik.

Random path representation and sharp correlations asymptotics at high-temperatures

We recently introduced a robust approach to the derivation of sharp asymptotic formula for correlation functions of statistical mechanics models in the high-temperature regime. We describe its application to the nonperturbative proof of Ornstein-Zernike asymptotics of 2-point functions for self-avoiding walks, Bernoulli percolation and ferromagnetic Ising models. We then extend the proof, in the Ising case, to arbitrary odd-odd correlation functions. We discuss the fluctuations of connection paths (invariance principle), and relate the variance of the limiting process to the geometry of the eq…

平均场Gibbs-Ornstein-Uhlenbeck过程的混合时间

The mixing time of Mean-Field Gibbs-Ornstein-Uhlenbeck processes

该论文提出平均场Gibbs-Ornstein-Uhlenbeck过程，证明其混合时间为n log n量级，并给出谱常数，统一了多种采样器与自旋模型的截止现象。

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。

On the two-point function of the Ising model with infinite range-interactions

In this article, we prove some results concerning the truncated two-point function of the infinite-range Ising model above and below the critical temperature. More precisely, if the coupling constants are of the form $J_{x}= ψ(x)e^{ -ρ(x)}$ with $ρ$ some norm and $ψ$ an subexponential correction, we show under appropriate assumptions that given $s\in\mathbb{S}^{d-1}$, the Laplace transform of the two-point function in the direction $s$ is infinite for $β=β_{\text{sat}}(s)$ (where $β_{\text{sat}}(s)$ is a the biggest value such that the inverse correlation length $ν_β(s)$ associated to the trun…