  In this article, we consider an anisotropic finite-range bond percolation
model on $\mathbb{Z}^2$. On each horizontal layer $\{(x,i): x \in \mathbb{Z}\}$
we have edges $\langle(x, i),(y, i)\rangle$ for $1 \leq |x - y| \leq N$. There
are also vertical edges connecting two nearest neighbor vertices on distinct
layers $\langle(x, i), (x, i+1)\rangle$ for $x, i \in\mathbb{Z}$. On this graph
we consider the following anisotropic independent percolation model: horizontal
edges are open with probability $1/(2N)$, while vertical edges are open with
probability $\epsilon$ to be suitably tuned as $N$ grows to infinity. The main
result tells that if $\epsilon=\kappa N^{-2/5}$, we see a phase transition in
$\kappa$: positive and finite constants $C_1, C_2$ exist so that there is no
percolation if $\kappa < C_1$ while percolation occurs for $\kappa > C_2$. The
question is motivated by a result on the analogous layered ferromagnetic Ising
model at mean field critical temperature [J. Stat. Phys. (2015), 161, 91-123]
where the authors showed the existence of multiple Gibbs measures for a fixed
value of the vertical interaction and conjectured a change of behavior in
$\kappa$ when the vertical interaction suitably vanishes as $\kappa\gamma^b$,
where $1/\gamma$ is the range of the horizontal interaction. For the product
percolation model we have a value of $b$ that differs from what was conjectured
in that paper. The proof relies on the analysis of the scaling limit of the
critical branching random walk that dominates the growth process restricted to
each horizontal layer and a careful analysis of the true horizontal growth
process. This is inspired by works on the long range contact process [Probab.
Th. Rel. Fields (1995), 102, 519-545]. A renormalization scheme is used for the
percolative regime.
