随机场随机表面热力学极限的收敛性

IF 1.4 2区 数学 Q2 STATISTICS & PROBABILITY Annals of Applied Probability Pub Date : 2021-05-09 DOI:10.1214/22-aap1844
P. Dario
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引用次数: 3

摘要

我们研究了在以随机独立外场形式存在猝灭无序的情况下,具有均匀凸梯度相互作用的随机表面。先前对该模型的工作集中于证明给定倾斜下无限体积梯度吉布斯测度的存在性和唯一性,以及研究表面及其离散梯度的波动。在这项工作中,我们专注于热力学极限的收敛性,建立了具有Dirichlet边界条件的有限体积分布到平移协变(梯度)Gibbs测度的收敛性。具体地说,当随机场律具有有限的二阶矩并且是对称的时,表面的梯度分布在维数$d\geq4$中收敛,而表面本身的分布在维数$d\geq 5$中收敛。此外,还得到了Wasserstein距离收敛速度的幂律上界。该结果部分回答了Cotar和K“ulske讨论的一个问题
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Convergence to the thermodynamic limit for random-field random surfaces
We study random surfaces with a uniformly convex gradient interaction in the presence of quenched disorder taking the form of a random independent external field. Previous work on the model has focused on proving existence and uniqueness of infinite-volume gradient Gibbs measures with a given tilt and on studying the fluctuations of the surface and its discrete gradient. In this work we focus on the convergence of the thermodynamic limit, establishing convergence of the finite-volume distributions with Dirichlet boundary conditions to translation-covariant (gradient) Gibbs measures. Specifically, it is shown that, when the law of the random field has finite second moment and is symmetric, the distribution of the gradient of the surface converges in dimensions $d\geq4$ while the distribution of the surface itself converges in dimensions $d\geq 5$. Moreover, a power-law upper bound on the rate of convergence in Wasserstein distance is obtained. The results partially answer a question discussed by Cotar and K\"ulske
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来源期刊
Annals of Applied Probability
Annals of Applied Probability 数学-统计学与概率论
CiteScore
2.70
自引率
5.60%
发文量
108
审稿时长
6-12 weeks
期刊介绍: The Annals of Applied Probability aims to publish research of the highest quality reflecting the varied facets of contemporary Applied Probability. Primary emphasis is placed on importance and originality.
期刊最新文献
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