Critical Percolation: the Expected Number of Clusters in a Rectangle

,
2011
Article

Résumé

We show that for critical site percolation on the triangular lattice two new observables have conformally invariant scaling limits. In par- ticular the expected number of clusters separating two pairs of points converges to an explicit conformal invariant. Our proof is independent of earlier results and SLE techniques, and in principle should provide a new approach to establishing conformal invariance of percolation

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Critical values in an anisotropic percolation on $\mathbb{Z}^2$

Hao Xue

In this thesis, we consider an anisotropic finite-range bond percolation model on

\mathbb{Z}^2

. On each horizontal layer

\{(x,i):x\in\mathbb{Z}\}

for

i\in\mathbb{Z}

, we have edges

\langle(x,i),(y,i)\rangle

for

1\leq|x-y|\leq N

with $N\in\mathbb{N ...

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