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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 '10 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 '10 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 '10 at 20:42
James, in the paper, Mean-field critical behaviour for percolation in high dimensions,… 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 '10 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 '10 at 1:26
up vote 4 down vote accepted

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:

Physicists have predicted some exponents describing asymptotics of critical percolation in 2D.

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 $$R^{−1/3+o(1)}, \qquad R \to \infty.$$

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In the case of oriented percolation the following regarding your question are rigorously known in any dimension. Presumably these results are also known for non-oriented percolation in half-spaces because in this case it is also known that there is no infinite cluster at criticality (Barsky, Grimmett and Newman 1991) and furthermore similar arguments apply for a continuous time percolation model where there is no orientation (i.e. time axis, see, Bezuidenhout and Grimmett 1991).

Let $\theta(p) = P(|C| = \infty)$, where $C$ is the cluster of the origin in oriented percolation with supercritical retention parameter $p > p_{c}$. Further, let $\theta(p; \gamma) = 1 - E_{p}(e^{\gamma |C|})$, $\gamma>0$. It is known that (Aizenman and Barsky 1987, Bezuidenhout and Grimmett 1991 or Aizenman and Jung 1991) there exist $a,b>0$ such that \begin{equation} \theta(p; \gamma) \geq a \gamma^{1 / 2}, \mbox{ for } 0<\gamma<b \end{equation} By means of emulating Tauberian theory arguments (due to Aizenmann and Barsky (1989), see Grimmett's v. 1991 book, Proposition 10.29) and using that $\theta(p_{c}) = 0$ (Bezuidenhout and Grimmett (1990)), the last display above is tantamount to \begin{equation} P( |C| \geq m) \approx m^{-\frac{1}{\delta}},\mbox{ } \delta \geq 2, \end{equation} where $\approx$ is meant in the logarithmic sence here ($a_{m} \approx b_{m}$ denotes $\frac{\log a_{m}}{\log b_{m}} \rightarrow 1$, as $m \rightarrow \infty$). Note that this implies that $P( |C| \geq m) \geq m^{-1 / 2}$, for all $m$ sufficiently large. In dimension 2 in particular the critical exponent for the probability of $\{0 \leftrightarrow \partial B(n)\}$, that is the origin being connected to the boundary of a box of size $n$, is furthermore known. Durrett (1988) proves that $\lim_{n \rightarrow \infty} \sqrt{n}P(0 \leftrightarrow \partial B(n)) = +\infty$. Hope this helps.

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