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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?

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  • $\begingroup$ 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. $\endgroup$
    – Timothy Budd
    Commented Feb 3, 2019 at 19:19

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See Aizenman, M.; Burchard, A. Hölder regularity and dimension bounds for random curves. Duke Math. J. 99 (1999), no. 3, 419--453. https://arxiv.org/abs/math/9801027

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