A Russo Seymour Welsh Theorem for critical site percolation on \(\mathbb{Z}^2\)

TL;DR

This paper proves a Russo-Seymour-Welsh type theorem for critical site percolation on , extending known results to cases lacking self-duality, and provides an accessible proof of Kesten's box crossing theorem.

Contribution

It establishes the RSW theorem for  site percolation without self-duality and offers a self-contained, accessible proof of Kesten's box crossing result.

Findings

RSW theorem holds for  site percolation at criticality.

No infinite cluster exists at criticality in  site percolation.

Provides an accessible proof of Kesten's box crossing theorem.

Abstract

The Russo-Seymour-Welsh Theorem for Z^2 bond or T (triangular lattice) site percolation states that at criticality, for all fixed real {\lambda}, the probability of the existence of a horizontal occupied crossing of each rectangle with size n*{\lambda}n is not degenerated when n tends to infinity. Turning to site percolation on Z^2, where the self duality does not hold anymore, we prove that the analogue statement of the RSW Theorem will still be true in this case. The proof uses appropriate finite size criteria and a result of Kesten which allows us to extend existing crossings without losing too much probability. As a consequence, there is no infinite cluster at criticalty. Our object in this short paper is twofold. Our first goal is to give a proof of a RSW Theorem for Z 2 site percolation. Since the proof uses in an essential way a celebrated result by Kesten on the so called "box crossing", our second goal in this paper is to present a self-contained proof of this theorem in perhaps a more accessible language.