Percolation theory critical density simple proof! Is there a simpler proof for the existence of infinite connected component in 2D lattice (percolation theory) if the probability of connection exceeds a critical threshold? Currently, I am reading the proof by Harry Kesten here. I am totally lost as I do not have a background to deal with that level of mathematics. I am hoping over the past 40 years someone might have come up with an easier proof.
Any help is appreciated!
 A: To provide a little bit of background for others: Consider the two-dimensional integer lattice $\mathbb{Z}^2$.  In bond percolation, edge of the lattice is either open or closed.  In particular, each edge is open with some probability $p$, independent of all other edges.
An open cluster is a component of $\mathbb{Z}^2$, all of whose edges are open.  A fundamental question in percolation theory asks, "For which $p$ is it true that the origin is contained in an infinite open cluster?"  Let $\theta(p)$ denote the event that the origin is in an infinite open cluster.  The critical probability is
$$p_c = p_c(\mathbb{Z}^2) = \inf\{p \in [0, 1] : \theta(p) > 0\}.$$
It was long conjectured that $p_c = 1/2$.  In 1960, Harris proved that $p_c \geq 1/2$.  In 1980, Kesten settled the conjecture by showing that $p_c \leq 1/2$.  (The proof that $p_c \leq 1/2$ is much harder than the proof that $p_c \geq 1/2$.)
B. Bollobas and O. Riordan gave a relatively simple proof that $p_c = 1/2$ in "A short proof of the Harris-Kesten Theorem" (https://arxiv.org/pdf/math/0410359.pdf).  The main simplification is in the proof of Kesten's theorem, but they also give a proof of Harris's theorem that is shorter than Harris's own.
