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A Percolation Formula

2001; Institute of Mathematical Statistics; Volume: 6; Issue: none Linguagem: Inglês

10.1214/ecp.v6-1041

1083-589X

Oded Schramm,

Mathematical Dynamics and Fractals

Let $A$ be an arc on the boundary of the unit disk $U$. We prove an asymptotic formula for the probability that there is a percolation cluster $K$ for critical site percolation on the triangular grid in $U$ which intersects $A$ and such that $0$ is surrounded by the union of $K$ and $A$.