Hydrodynamic Limit of Zero Range Processes Among Random Conductances on the Supercritical Percolation Cluster
Hydrodynamic Limit of Zero Range Processes Among Random Conductances on the Supercritical Percolation Cluster
复制标题
超临界渗流团簇随机电导零范围过程的水动力极限
DOI：
10.1214/ejp.v15-748
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发表时间：
2008
影响因子：
1.4
通讯作者：
A. Faggionato
中科院分区：
文献类型：
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作者：
A. Faggionato
我们考虑 i.i.d.随机变量 $\omega=\{\omega(b)\}$ 由 $\mathbb{Z}^d$ 中的键族参数化，$d > 1$。随机变量 $\omega(b)$ 被认为是键 $b$ 的电导，它的范围在有限区间 $[0,c_0]$ 内。假设事件 $\{\omega(b) > 0\}$ 的概率为超临界，并用 $C(\omega)$ 表示与正电导键相关的唯一无限簇，我们研究 $C(\omega)$ 上的零范围过程，其中 $\omega(b)$ 是沿键 $b$ 跳跃的比例概率率。对于几乎所有的环境实现，我们证明零范围过程的流体动力学行为是由非线性热方程控制的，与 $\omega$ 无关。作为上述结果和有限簇的阻塞效应的副产品，我们讨论具有电导场 $\omega$ 的 $\mathbb{Z}^d$ 上零范围过程的体行为。我们不需要任何椭圆度条件。
We consider i.i.d. random variables $\omega=\{\omega(b)\}$ parameterized by the family of bonds in $\mathbb{Z}^d$, $d > 1$. 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 the probability of the event $\{\omega(b) > 0\}$ to be supercritical and denoting by $C(\omega)$ the unique infinite cluster associated to the bonds with positive conductance, we study the zero range process on $C(\omega)$ with $\omega(b)$-proportional probability rate of jumps along bond $b$. For almost all realizations of the environment we prove that the hydrodynamic behavior of the zero range process is governed by a nonlinear heat equation, independent from $\omega$. As byproduct of the above result and the blocking effect of the finite clusters, we discuss the bulk behavior of the zero range process on $\mathbb{Z}^d$ with conductance field $\omega$. We do not require any ellipticity condition.