  We consider the following 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 a fixed $\epsilon$ between 0 and 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 edge set $\mathbb
E = \{(a, a \pm e_i): a \in \mathbb Z^2, \, i = 1, 2\}$, where $e_1$ and $e_2$
are the unit coordinate vectors. We define notions of weak and strong
percolation. When $\epsilon = 0$ the model fails to percolate weakly, and when
$\epsilon = 1$, 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 establish that in general, the number of infinite
strong clusters is 0, 1, or $\infty$ almost surely, and some other related
results.
