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## half-plane percolation clusters

Consider critical edge-percolation in the induced subgraph of the square grid with vertex set {$(i,j) \in Z \times Z:\ i+j \geq 0$}, and let $p_n$ be the probability that the cluster containing $(0,0)$ has size $> n$. How quickly does $p_n$ fall as $n \rightarrow \infty$?

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Very nice question. Are there reasons to believe that the behavior for the half plane is different than the behavior for the full plane? – Gil Kalai May 2 2010 at 4:54
I have no reason to think that the behavior for the half plane is different than the behavior for the full plane, so I guess I should ask, by way of background, what the answer is for percolation in the full plane. – James Propp May 2 2010 at 5:15
James, I agree with Gil in that it should probably be the same qualitative phenomenon as the full lattice. Of course, the critical percolation probability will no longer be 1/2 as on the full square lattice. What's your motivation for studying this question? – Tom LaGatta May 2 2010 at 20:42
James, in the paper, Mean-field critical behaviour for percolation in high dimensions, math.bme.hu/~balint/oktatas/perkolacio/… Hara and Slade mention on the page 337 that $$\mathbb P_{p_c}(\{|C(0)|=n\})\sim n^{-1-1/\delta}.$$ I did not remember if this asymptotic behavior it is expected only for high dimensions and if the invariance properties of $p_{uv}$ are required, anyway there is some references about this problem in the same page, maybe you can find something relevant for your problem there. – Leandro May 2 2010 at 21:43
Thanks to Gil, Tom, and Leandro for their comments. I don't understand Tom's remark that $p_{\rm crit}$ will no longer be 1/2 for the diagonal half-plane model; can Tom (or someone else) explain this? In any case, when I wrote "critical edge-percolation" in my original posting, I meant that each edge is included with probability 1/2. The Nienhuis and Cardy conjectures, taken together, seem to suggest that the answer to Gil's question ("Are there reasons to believe that the behavior for the half plane is different than the behavior for the full plane?") is an emphatic "Yes". – James Propp May 4 2010 at 1:26
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As Leandro suggested in the comments, this should follow a power-law decay in $n$. However, Hara and Slade's rigorous work using lace expansions is only valid for dimensions $\ge 19$. Much of the rigorous work on critical exponents for two-dimensional percolation has been done only recently, via connections to Schramm-Loewner Evolution (with $\kappa = 6$). A good starting place might be this PowerPoint presentation by Oded Schramm. Here is Page 23:
For example, Nienhuis conjectured that the probability that the origin is in a cluster of diameter $\ge R$ is $$R^{−5/48+o(1)}, \qquad R \to \infty$$ and Cardy conjectured that the probability that the origin is connected to distance $R$ within the upper half plane is $$R^{−1/3+o(1)}, \qquad R \to \infty.$$