# What is the expected distance between the sides of a random subgraph of the grid?

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?

• The right keyword seems to be "chemical distance". Perhaps the following references are relevant: Antal, P., & Pisztora, A. (1996). On the Chemical Distance for Supercritical Bernoulli Percolation. The Ann. Prob., 24(2), 1036-1048, and Grimmett, G. R., & Marstrand, J. M. The supercritical phase of percolation is well behaved. Proc. R. Soc. Lond. A, 430(1879), 439-457. – Timothy Budd Feb 3 at 19:19