# Why, and how badly, does the proof of “no percolation at the critical point in half-spaces” fail for full spaces?

The proof by Barsky et. al. that there is no percolation in half-spaces proceeds by a dynamic renormalization argument. The proof couples critical percolation in the half-space $\mathbb{H}^d$ with a dependant site percolation model on $\mathbb{Z}^2$ such that if the block sizes in the renormalization are sufficiently large then the dependant percolation on $\mathbb{Z}^2$ is supercritical. Since the block size is still finite, the probability that a block is "good" is a polynomial in $p$ (the edge probability), and a continuity argument can then be used to show that if $\Theta(p_c)>0$ (for $\mathbb{H}^d$) then there is in fact $p < p_c$ for which the dependent percolation on $\mathbb{Z}^2$ is still supercritical, and so the half-space in fact already contained an infinite component at $p$.

While in principle I understand this style of proof, I realized that I don't understand this specific argument well enough to know just what goes wrong if we try to run the same argument on $\mathbb{Z}^d$ rather than $\mathbb{H}^d$. In fact, I don't even understand whether the reason it breaks down essentially technical, or whether there are good reasons to believe that the same line of attack is very unlikely to work for $\mathbb{Z}^d$. (One good reason to have the latter belief is that lots of smart people have thought about the problem without success, but that's not the kind of reason I mean.)

So: Why, and how badly, does the proof of "no percolation in half-spaces" fail for full spaces?

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