pr.probability - half-plane percolation clusters

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.