Quenched Large Deviations for Simple Random Walks on Percolation Clusters Including Long-Range Correlations
Quenched Large Deviations for Simple Random Walks on Percolation Clusters Including Long-Range Correlations
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DOI：
10.1007/s00220-017-3054-z
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发表时间：
2016-12
影响因子：
2.4
通讯作者：
Noam Berger;Chiranjib Mukherjee;K. Okamura
中科院分区：
文献类型：
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作者：
Noam Berger;Chiranjib Mukherjee;K. Okamura
本文证明了（）上超临界渗流团簇（SRWPC）上简单随机游动的量子化大偏差原理（LDP）。感兴趣的模型包括经典的伯努利键和网站渗流以及模型，表现出长程相关性，如随机簇模型，随机交织和随机交织的空集（为）和高斯自由场的水平集（）。受Kosygina等人开发的方法的启发。（Commun Pure Appl Math 59：1489-1521，2006），以及Yilmaz（Commun Pure Appl Math 62（8）：1033-1075，2009）和Rosenbluth（Quenched large deviations for multidimensional random walks in a random environment：a variation formula.博士论文，纽约大学， arXiv：0804.1444v1 对于随机环境中椭圆随机游动的类似结果，我们从运动粒子的角度证明了非椭圆情形下SRWPC环境马氏链的经验测度对的猝灭分布的大偏差原理。通过收缩原理，这很容易减少到淬火LDP的随机行走的平均速度的分布和速率函数承认明确的变分公式。在我们的设置中的主要困难在于固有的非椭圆性，以及缺乏fractionation-invariancesstemming条件下的事实，即起源属于无限集群。我们开发了一个统一的方法来证明淬火大偏差SRWPC的基础上利用凸变分分析的上下文中的相对熵的avivity属性，结合输入遍历理论和调用的超临界渗流集群的几何性质。
We prove aquenchedlarge deviation principle (LDP) for a simple random walk on a supercritical percolation cluster (SRWPC) on(). The models under interest include classical Bernoulli bond and site percolation as well as models that exhibit long range correlations, like the random cluster model, the random interlacement and the vacant set of random interlacements (for) and the level sets of the Gaussian free field (). Inspired by the methods developed by Kosygina et al. (Commun Pure Appl Math 59:1489–1521, 2006) for proving quenched LDP for elliptic diffusions with a random drift, and by Yilmaz (Commun Pure Appl Math 62(8):1033–1075, 2009) and Rosenbluth (Quenched large deviations for multidimensional random walks in a random environment: a variational formula. Ph.D. thesis, NYU, arXiv:0804.1444v1 ) for similar results regarding elliptic random walks in random environment, we take the point of view of the moving particle and prove a large deviation principle for the quenched distribution of thepair empirical measuresof the environment Markov chain in the non-elliptic case of SRWPC. Via a contraction principle, this reduces easily to a quenched LDP for the distribution of the mean velocity of the random walk and both rate functions admit explicit variational formulas. The main difficulty in our set up lies in the inherent non-ellipticity as well as the lack oftranslation-invariancestemming from conditioning on the fact that the origin belongs to the infinite cluster. We develop a unifying approach for proving quenched large deviations for SRWPC based on exploiting coercivity properties of the relative entropies in the context of convex variational analysis, combined with input from ergodic theory and invoking geometric properties of the supercritical percolation cluster.