Poincaré inequality for Markov random fields via disagreement percolation

Random walks on infinite percolation clusters in models with long-range correlations

For a general class of percolation models with long-range correlations on $\mathbb Z^d$, $d\geq 2$, introduced in arXiv:1212.2885, we establish regularity conditions of Barlow arXiv:math/0302004 that mesoscopic subballs of all large enough balls in the unique infinite percolation cluster have regular volume growth and satisfy a weak Poincaré inequality. As immediate corollaries, we deduce quenched heat kernel bounds, parabolic Harnack inequality, and finiteness of the dimension of harmonic functions with at most polynomial growth. Heat kernel bounds and the quenched invariance principle of arX…

Exponential decay of correlations in the 2D random field Ising model

An extension of the Ising spin configurations to continuous functions is used for an exact representation of the Random Field Ising Model's order parameter in terms of disagreement percolation. This facilitates an extension of the recent analyses of the decay of correlations to positive temperatures, at homogeneous but arbitrarily weak disorder.

通过分歧耦合对吉布斯过程的正态近似

Normal approximation for Gibbs processes via disagreement couplings

本文通过扩展分歧耦合方法，在弱稳定化、渗流阈值范围内的相互作用以及多 Palm 过程同时耦合三个方面改进了吉布斯过程几何泛函的中心极限定理，并给出柯尔莫哥洛夫界形式的定量正态近似。

A simplified second-order Gaussian Poincaré inequality in discrete setting with applications

In this paper, a simplified second-order Gaussian Poincaré inequality for normal approximation of functionals over infinitely many Rademacher random variables is derived. It is based on a new bound for the Kolmogorov distance between a general Rademacher functional and a Gaussian random variable, which is established by means of the discrete Malliavin-Stein method and is of independent interest. As an application, the number of vertices with prescribed degree and the subgraph counting statistic in the Erdös-Rényi random graph are discussed. The number of vertices of fixed degree is also studie…

利用随机游走和渗流为高斯自由场生成Galton-Watson树

Generating Galton-Watson trees using random walks and percolation for the Gaussian free field

本文通过有限随机游走轨迹探索构造Galton-Watson树，结合Dynkin型同构定理，将高斯自由场渗流临界参数$h_*$严格为正的结果推广到任意超临界后代分布和随机电导情形，并证明正相关促进渗流。

Gibbsianness of locally thinned random fields

We consider the locally thinned Bernoulli field on $\mathbb Z^d$, which is the lattice version of the Type-I Matérn hardcore process in Euclidean space. It is given as the lattice field of occupation variables, obtained as image of an i.i.d. Bernoulli lattice field with occupation probability $p$, under the map which removes all particles with neighbors, while keeping the isolated particles. We prove that the thinned measure has a Gibbsian representation and provide control on its quasilocal dependence, both in the regime of small $p$, but also in the regime of large $p$, where the thinning tr…

Quenched invariance principle for a long-range random walk with unbounded conductances

We consider a random walk on a random graph $(V,E)$, where $V$ is the set of open sites under i.i.d. Bernoulli site percolation on the multi-dimensional integer set $\mathbf{Z}^d$, and the transition probabilities of the walk are generated by i.i.d. random conductances (positive numbers) assigned to the edges in $E$. This random walk in random environments has long range jumps and is reversible. We prove the quenched invariance principle for this walk when the random conductances are unbounded from above but uniformly bounded from zero by taking the corrector approach. To this end, we prove a …

随机场Ising动力学的快速弛豫

Fast relaxation of the random field Ising dynamics

研究了随机场Ising模型Glauber动力学的收敛性，在弱和强空间混合条件下分别证明了多项式和近多项式时间的混合，并构建了相应的近似采样算法。