Let $G$ be the $n \times n$ grid, in which each vertex is connected to the vertices above it, below it and on either side. Let $G_p$ be the random subgraph of $G$ obtained by keeping each edge with probability $p$. A lot of research has focused on determining $p_C$, the critical threshold probability of having a connected component of size $\Omega(n^2)$.
I am interested in a related problem. Define the width of a subgraph of the grid to be the minimal distance between a vertex on the left side and a vertex on the right side of the grid.
Now, let $p>p_C$. What can be said about the expected width of $G_p$?
I am a newcomer to the field, so I am looking for some expert advice: Is this problem solved, known and open, or neither?