Consider the following first-passage percolation problem on the plane grid $\mathbb{Z}^2$: all the horizontal edges are directed (pointing east) and carry an independent random weight, say standard exponential. All the vertical edges are undirected and carry a deterministic weight, say $c$.

As usual, the length of a path between two vertices is the sum of the weights of its edges. I am interested in the geometry of the shortest path between the origin and the point $(n,0)$. I am particular interested in the limit $n\to\infty$ when $c=c_n$ is allowed to vary (get smaller) with $n$

**Question**: Has this model been studied before? What are recommended references to start reading about it?

Below is a picture for $n=100$ and $c=0.05$;

**Edit**:
The specification that the horizontal edges are directed is superfluous. Having undirected horizontal edges results in the same model.