Consider critical edgepercolation 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 $n \rightarrow \infty$?

As Leandro suggested in the comments, this should follow a powerlaw 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 twodimensional percolation has been done only recently, via connections to SchrammLoewner 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. 


In the case of oriented percolation the following regarding your question are rigorously known in any dimension. Presumably these results are also known for nonoriented percolation in halfspaces 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. 

