标题： 随机电阻网络在简单点过程上的电导率的尺度极限 显示英文标题

标题： Scaling limit of the conductivity of random resistor networks on simple point processes

摘要： 我们考虑节点由欧式空间$\mathbb{R}^d$上的简单点过程给出的随机电阻网络，并具有随机电导率。 电流细丝的长度范围可以是无界的。 我们假设随机性相对于群$\mathbb{G}$的作用是平稳且遍历的，该群作用由$\mathbb{R}^d$或$\mathbb{Z}^d$给出。 该作用相对于欧式空间上的平移是协变的。 在最小假设下，我们证明几乎必然地，电阻网络沿有效均质矩阵$D$主方向的适当缩放后的方向电导率收敛于$D$的对应特征值乘以简单点过程的强度。 我们还通过适当修改物理观测区域的形状，证明了任何与$D$核正交的方向上的电导率的几乎必然尺度极限。 我们的结果涵盖了大量模型，包括例如 标准导电模型在$\mathbb{Z}^d$上，Miller-Abrahams 无定形固体导电的电阻网络（我们现在可以将其范围扩展至与 Mott 的定律一致，此前在\cite{CP1,FM,FSS}中获得的 Mott 随机游走的结果），在格子和连续渗流的超临界团簇上的电阻网络，晶体格子和 Delaunay 三角剖分上的电阻网络。 显示英文摘要

摘要： We consider random resistor networks with nodes given by a simple point process on $\mathbb{R}^d$ and with random conductances. The length range of the electrical filaments can be unbounded. We assume that the randomness is stationary and ergodic w.r.t.~the action of the group $\mathbb{G}$, given by $\mathbb{R}^d$ or $\mathbb{Z}^d$. This action is covariant w.r.t.~ translations on the Euclidean space. Under minimal assumptions we prove that a.s. the suitably rescaled directional conductivity of the resistor network along the principal directions of the effective homogenized matrix $D$ converges to the corresponding eigenvalue of $D$ times the intensity of the simple point process. We also prove the a.s. scaling limit of the conductivity along any direction perpendicular to the kernel of $D$ by suitably modifying the shape of the region of physical observation. Our results cover plenty of models including e.g. the standard conductance model on $\mathbb{Z}^d$, the Miller-Abrahams resistor network for conduction in amorphous solids (to which we can now extend the bounds in agreement with Mott's law previously obtained in \cite{CP1,FM,FSS} for Mott's random walk), resistor networks on the supercritical cluster in lattice and continuum percolations, resistor networks on crystal lattices and on Delaunay triangulations.