Why, and how badly, does the proof of "no percolation at the critical point in half-spaces" fail for full spaces? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T23:14:29Z http://mathoverflow.net/feeds/question/38336 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/38336/why-and-how-badly-does-the-proof-of-no-percolation-at-the-critical-point-in-ha Why, and how badly, does the proof of "no percolation at the critical point in half-spaces" fail for full spaces? Louigi Addario-Berry 2010-09-10T17:35:50Z 2010-09-10T17:46:38Z <p>The proof by <a href="http://www.springerlink.com/content/m18532022p572078/" rel="nofollow">Barsky et. al.</a> that there is no percolation in half-spaces proceeds by a dynamic renormalization argument. The proof couples <em>critical</em> 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 &lt; p_c\$ for which the dependent percolation on \$\mathbb{Z}^2\$ is <em>still</em> supercritical, and so the half-space in fact already contained an infinite component at \$p\$. </p> <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.) </p> <blockquote> <p>So: Why, and how badly, does the proof of "no percolation in half-spaces" fail for full spaces?</p> </blockquote>