Consider classical bond percolation on $\mathbb{Z}^d$. Each edge is included with probability $p$ and deleted with probability $1-p$. As is well known, there is a $p_c(d) \in (0,1)$ such that $p>p_c$ means there is an infinite connected component of the resulting graph (with probability one), and $p<p_c$ means there is no infinite connected component.
We are now going to play a game in which $p$ is fixed, but you are not told the value of $p$. You do however know that $p \not \in (p_c -\delta, p_c+\delta)$ for some $\delta>0$ which is revealed to you.
You are allowed to take ONE sample of a cube of size $N$. You are not allowed to look at the sample, all you are allowed to know is whether there exists a connected component of the graph in the cube which touches all the boundary faces of the cube.
That is, you are given one (1) binary bit of information.
My question is: Given a small $\epsilon>0$, How large does $N$ need to be, as a function of $(\epsilon,\delta)$, for you to correctly guess, based on this one binary bit of inforation, with probability at least $1-\epsilon$, whether $p>p_c$ or $p<p_c$.
I am really interested in quantitative information which is asymptotic in $\delta$. You can simplify by taking $\epsilon = 0.05$ if you like.
I would like to have all constants determined explicitly (in principle). That is, whatever theorem or paper is being quoted should not, preferably, have a "qualitative step" in which information about $\delta$ dependence is lost.
I am expecting that a completely satisfactory answer does not exist, so here are some sub-questions.
Are there any upper bounds on $N$ whatsoever in general dimension?
Are there specific examples of lattices (perhaps the 2d hexagonal lattice?) in which we know a complete answer? Do we know anything about the square lattice $\mathbb{Z}^2$?
(Also, if you don't like my binary bit of information and you want to replace it by some other yes/no question about a single sample in a box of size $N$, that is ok.)