Random walks and exclusion processes among random conductances on random infinite clusters: homogenization and hydrodynamic limit
Random walks and exclusion processes among random conductances on random infinite clusters: homogenization and hydrodynamic limit
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随机无限簇上随机电导之间的随机游走和排除过程：均质化和流体动力学极限
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发表时间：
2007
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影响因子：
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通讯作者：
A. Faggionato
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作者：
A. Faggionato
本文考虑一个平稳遍历的随机场${omega（B）：B in mathbb{E}_d }$由$mathbb{Z}^d$，$dgeq 2$中的键族参数化.随机变量$omega（B）$被认为是键$B$的电导，它在有限区间$[0，c_0]$内变化。假设具有正电导的键的集合具有唯一的无限簇$mathcal{C}（omega）$，我们证明了$mathcal{C}（omega）$上随机电导之间的随机游走的均匀化结果。作为一个副产品，应用Faggionato（2007）的一般标准，导致排斥过程的流体动力学极限与键-依赖的过渡率，几乎所有的实现环境中，我们证明了简单的排斥过程的流体动力学极限之间的随机电导$mathcal{C}（ω）$。流体动力学方程由热方程给出，其扩散矩阵不依赖于环境。我们不需要任何椭圆性条件。作为特殊情况，$mathcal{C}（omega）$可以是超临界伯努利键渗流的无限簇。
We consider a stationary and ergodic random field ${omega (b):b in mathbb{E}_d }$ parameterized by the family of bonds in $mathbb{Z}^d$, $dgeq 2$. The random variable $omega(b)$ is thought of as the conductance of bond $b$ and it ranges in a finite interval $[0,c_0]$. Assuming that the set of bonds with positive conductance has a unique infinite cluster $mathcal{C}(omega)$, we prove homogenization results for the random walk among random conductances on $mathcal{C}(omega)$. As a byproduct, applying the general criterion of Faggionato (2007) leading to the hydrodynamic limit of exclusion processes with bond--dependent transition rates, for almost all realizations of the environment we prove the hydrodynamic limit of simple exclusion processes among random conductances on $mathcal{C}(omega)$. The hydrodynamic equation is given by a heat equation whose diffusion matrix does not depend on the environment. We do not require any ellipticity condition. As special case, $mathcal{C}(omega)$ can be the infinite cluster of supercritical Bernoulli bond percolation.