Site Percolation and S-clusters with $n\times m$ grid where $n\not =m$?

Question

Consider a site percolation but change the dimension of the lattice from $n\times n$ to $n\times m$ where $n\not = m.$ S-clusters are defined for $n\times n$ lattice. The occupation of each site are independent of each other.

Example about site percolation with $m=n$, a square lattice.

Example about site percolation where $m\not =n$, non-square lattice.

Is the non-square model still called site percolation?

P.s. Super interesting site percolation programming here for the shortest path.

Answer 1

You want to read on Russo-Seymour-Welsh (RSW) theorem:

"The Russo-Seymour-Welsh (RSW) theory is one of the most important tools in the study of planar percolation. A RSW-result generally refers to an inequality that provides a bound on the probability to cross rectangles in the long direction" Source.

and Antoined P:

"This was a key to prove the conformal invariance of the site hexagonal lattice. Have a look at the lecture notes of Hugo Duminil-Copin or Ariel Yadin on the matter."

• Ariel Yadin in Arxiv and his lectures

• Hugo Duminil-Copin in Arxiv and Hugo's lectures

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RSW Theorem

• https://www.unige.ch/~duminil/publi/2015IAMPproceedings.pdf

• http://www.unige.ch/~tassion/fichiersPDF/voronoi.pdf