Harness processes and harmonic crystals

Harness Processes and Non-Homogeneous Crystals

We consider the Harmonic crystal, a measure on $\mathbb{R}^{\mathbb{Z}^{d}}$ with Hamiltonian $H(\x)=\sum_{i,j}J_{i,j}(\x(i)-\x(j))^{2}+ h\sum_{i}(\x(i)-\dd(i))^{2}$, where $\x, \dd$ are configurations, $\x(i),\dd(i)\in\mathbb{R}$, $i,j\in{\mathbb{Z}^{d}}$. The configuration $\dd$ is given and considered as observations. The `couplings' $J_{i,j}$ are finite range. We use a version of the harness process to explicitly construct the unique infinite volume measure at finite temperature and to find the unique ground state configuration $\m$ corresponding to the Hamiltonian.

On the convergence to statistical equilibrium for harmonic crystals

We consider the dynamics of a harmonic crystal in $d$ dimensions with $n$ components, $d,n$ arbitrary, $d,n\ge 1$, and study the distribution $μ_t$ of the solution at time $t\in\R$. The initial measure $μ_0$ has a translation-invariant correlation matrix, zero mean, and finite mean energy density. It also satisfies a Rosenblatt- resp. Ibragimov-Linnik type mixing condition. The main result is the convergence of $μ_t$ to a Gaussian measure as $t\to\infty$. The proof is based on the long time asymptotics of the Green's function and on Bernstein's ``room-corridors'' method.

On two-temperature problem for harmonic crystals

We consider the dynamics of a harmonic crystal in $d$ dimensions with $n$ components,$d,n \ge 1$. The initial date is a random function with finite mean density of the energy which also satisfies a Rosenblatt- or Ibragimov-Linnik-type mixing condition. The random function converges to different space-homogeneous processes as $x_d\to\pm\infty$, with the distributions $μ_\pm$. We study the distribution $μ_t$ of the solution at time $t\in\R$. The main result is the convergence of $μ_t$ to a Gaussian translation-invariant measure as $t\to\infty$. The proof is based on the long time asymptotics of …

Hammersley's harness process: invariant distributions and height fluctuations

We study the invariant distributions of Hammersley's serial harness process in all dimensions and height fluctuations in one dimension. Subject to mild moment assumptions there is essentially one unique invariant distribution, and all other invariant distributions are obtained by adding harmonic functions of the averaging kernel. We identify one Gaussian case where the invariant distribution is i.i.d. Height fluctuations in one dimension obey the stochastic heat equation with additive noise (Edwards-Wilkinson universality). We prove this for correlated initial data subject to polynomial decay …

二维谐振子的硬墙概率的二阶渐进行为

Second Order Asymptotics for the Hard Wall Probability of the 2D Harmonic Crystal

本文研究了二维谐振子在特定边界条件下的概率行为，通过改进已有结果，推导了该概率的次主导项阶数，并利用中心最小值的双指数衰减特性作为关键证明工具。

Note on Bolthausen-Deuschel-Zeitouni's paper on the absence of a wetting transition for a pinned harmonic crystal in dimensions three and larger

The article [Bolthausen et al., 2000] provides a proof of the absence of a wetting transition for the discrete Gaussian free field conditioned to stay positive, and undergoing a weak delta-pinning at height 0. The proof is generalized to the case of a square pinning-potential replacing the delta-pinning, but it relies on a lower bound on the probability for the field to stay above the support of the potential, the proof of which appears to be incorrect. We provide a modified proof of the absence of a wetting transition in the square-potential case, which does not require the aforementioned low…