half-plane percolation clusters

Question

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 $n \rightarrow \infty$?

Answer 1

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

Answer 2

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.