The Lightning Model

Abstract

We introduce a non-standard model for percolation on the integer lattice $\mathbb Z^2$. Randomly assign to each vertex $a \in \mathbb Z^2$ a potential, denoted $\phi_a$, chosen independently and uniformly from the interval $[0, 1]$. For fixed $\epsilon \in [0,1]$, draw a directed edge from vertex $a$ to a nearest-neighbor vertex $b$ if $\phi_b < \phi_a + \epsilon$, yielding a directed subgraph of the infinite directed graph $\overrightarrow{G}$ whose vertex set is $\mathbb Z^2$, with nearest-neighbor edge set. We define notions of weak and strong percolation for our model, and observe that when $\epsilon = 0$ the model fails to percolate weakly, while for $\epsilon = 1$ it percolates strongly. We show that there is a positive $\epsilon_0$ so that for $0 \le \epsilon \le \epsilon_0$, the model fails to percolate weakly, and that when $\epsilon > p_\text{site}$, the critical probability for standard site percolation in $\mathbb Z^2$, the model percolates strongly. We study the number of infinite strongly connected clusters occurring in a typical configuration. We show that for these models of percolation on directed graphs, there are some subtle issues that do not arise for undirected percolation. Although our model does not have the finite energy property, we are able to show that, as in the standard model, the number of infinite strongly connected clusters is almost surely 0, 1 or $\infty$.