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.


1. Are there any upper bounds on $N$ whatsoever in general dimension?
2. 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.)



Percolation: at what length scale do we see it?
As Ofer said in his comment, this is should be equivalent to estimating the correlation length. Of course there are a few different ways to define the correlation length, but usually proving these ways are equivalent can be done in a completely quantitative way. E.g. in his scaling relation paper <https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-109/issue-1/Scaling-relations-for-2D-percolation/cmp/1104116714.full> Kesten proves a "stability of crossing probability" characterisation of the correlation length in 2d which I think is exactly what you want, and it should be possible to extract explicit estimates on the constants from his paper if you really wanted to. In high dimensions box-crossing probabilities are typically not the most natural thing to look at and I'm not aware of anywhere you could extract a complete answer to your question just by proof-mining the relevant constants out (but you should get an answer up to a log factor by taking a union bound).


In general dimension, the best known bound is in this paper of Duminil-Copin, Kozma, and Tassion. <https://link.springer.com/chapter/10.1007/978-3-030-60754-8_16>, who prove that the correlation length is at most $e^{C\delta^{-2}}$ for some dimension-dependent constant $C$, which if I recall correctly can be computed-in-principle from their proof, and is probably not even too difficult to get in this case (one would first have to determine the universal constant in Talagrand's sharp threshold theorem). Of course this bound is way off what is expected to hold.


In high dimensions, these things are pretty well-understood (under appropriate quantitative conditions, e.g. assuming the dimension is very large) via the lace expansion, and these lace expansion proofs can always produce explicit estimates on constants if desired (indeed, one often needs to prove that some constant is very close to 1 for the method to work at all). In particular, the asymptotics of the subcritical correlation length were computed by Hara <https://link.springer.com/article/10.1007/BF01208256>, and (while I'm not very familiar with the details of his paper) I imagine you can extract explicit estimates on all the constants if you really want to. See also my paper <https://arxiv.org/abs/2107.12971> with Emmanuel Michta and Gordon Slade; If I recall correctly all our proofs are effective, but one would have to go back to the earlier papers we use and make sure the constants there can also be computed effectively (which seems like a daunting task since there are a lot of dependencies to prior work). I really doubt this would be a problem but it would probably be a huge pain. Of course if you want a box-crossing estimate you would also have to do some additional work, but just using a union bound over all points should only make things worse by a log factor so probably not a big deal for whatever computer-assisted proof you have in mind.


By the way, depending on exactly what you want to do it is probably better to use bounds that have better leading constants than do better estimating the right exponents. Usually we understand near-critical percolation by first understanding critical percolation. For the second step, going from critical to slightly subcritical, there are simple inequalities with reasonable constants available in all dimensions, but which are only sharp (in terms of exponents) in high dimensions, <https://arxiv.org/abs/1901.10363>.


In two dimensions, my understanding is that the exponent calculations for site percolation on the hexagonal lattice don't give effective bounds since they rely on a "soft" step where you use the convergence to SLE (i.e. you end up with quantities guaranteed to satisfy power laws up to inexplicit $o(1)$ terms in the exponents). If you just want a reasonable bound with the wrong exponents it shouldn't be too hard to get something from RSW (to get an explicit power-law upper bound on the one-arm probability) before turning this into a explicit exponential one-arm bound using the methods of my paper linked in the previous paragraph (but applied to the one-arm probability rather than the volume tail, where the relevant differential inequality is provided by e.g. <https://arxiv.org/abs/1705.03104>). I think it should be possible to do this in a fairly painless way and with constants that aren't too tiny/enormous.


PS. One reason people tend not to be interested in leading constants is that they are usually non-universal (i.e. lattice-dependent), so it's against the philosophy of the renormalization group, universality etc to care about them. An important exception is that leading constants often *are* universal for models at the upper-critical dimension (six for percolation), although even then we would of course still have that the rate of convergence to the relevant asymptotic formula is non-universal.


