half-plane percolation clusters

2010-05-02T04:45:04Z

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 by Tom LaGatta for half-plane percolation clusters

2010-05-02T23:00:30Z

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

half-plane percolation clusters - MathOverflow

half-plane percolation clusters

Answer by Tom LaGatta for half-plane percolation clusters