Pinning for the critical and supercritical membrane model

Fast decay of covariances under $δ-$pinning in the critical and supercritical membrane model

We consider the membrane model, that is the centered Gaussian field on $\mathbb Z^d$ whose covariance matrix is given by the inverse of the discrete Bilaplacian. We impose a $δ-$pinning condition, giving a reward of strength $\varepsilon$ for the field to be $0$ at any site of the lattice. In this paper we prove that in dimensions $d\geq 4$ covariances of the pinned field decay at least stretched-exponentially, as opposed to the field without pinning, where the decay is polynomial in $d\geq 5$ and logarithmic in $d=4.$ The proof is based on estimates for certain discrete Sobolev norms, and on …

Exponential decay of covariances for the supercritical membrane model

We consider the membrane model, that is the centered Gaussian field on $\mathbb Z^d$ whose covariance matrix is given by the inverse of the discrete Bilaplacian. We impose a $δ-$pinning condition, giving a reward of strength $\varepsilon$ for the field to be $0$ at any site of the lattice. In this paper we prove that in dimensions $d\geq 5$ covariances of the pinned field decay at least exponentially, as opposed to the field without pinning, where the decay is polynomial. The proof is based on estimates for certain discrete weighted norms, a percolation argument and on a Bernoulli domination r…

非临界半空间几何最后通过渗透中的固定点

Pinning in non-critical half-space geometric last passage percolation

本文研究了对称化（半空间）的几何最后通过渗透模型，通过边界参数c介于亚临界、临界和超临界行为之间。该模型产生了一族交错的随机曲线或线集，编码了通常的最后通过时间及其高阶类比。在亚临界（c < 1）和超临界（c > 1）阶段，确定了线集在近对角线区域的通用标度极限。

Extremes of some Gaussian random interfaces

In this article we give a general criterion for some dependent Gaussian models to belong to maximal domain of attraction of Gumbel, following an application of the Stein-Chen method studied in Arratia et al(1989). We also show the convergence of the associated point process. As an application, we show the conditions are satisfied by some of the well-known supercritical Gaussian interface models, namely, membrane model, massive and massless discrete Gaussian free field, fractional Gaussian free field.

铁磁伊辛模型的临界与近临界影响界

Critical and near-critical influence bounds for ferromagnetic Ising models

针对铁磁伊辛模型，证明影响矩阵在临界阈值处有根号n阶界，并推广到近临界情形，进而得出Glauber动力学在超临界窗口内多项式时间混合。