A simple proof of exponential decay in the two dimensional percolation model

TL;DR

This paper provides a simpler proof of Kesten's theorem, demonstrating exponential decay of percolation probability in the subcritical phase for 2D percolation models, confirming critical probabilities for certain lattices.

Contribution

It introduces a more straightforward proof of exponential decay in 2D percolation, simplifying Kesten's original argument and confirming critical probabilities.

Findings

Exponential decay of percolation probability in subcritical phase

Critical probabilities are 0.5 for specific lattices

Simplified proof technique for Kesten's theorem

Abstract

Kesten showed the exponential decay of percolation probability in the subcritical phase for the two-dimensional percolation model. This result implies his celebrated computation that $p_c=0.5$ for bond percolation in the square lattice, and site percolation in the triangular lattice, respectively. In this paper, we present a simpler proof for Kesten's theorem.