Edge percolation on Z2: percolation probability of every point is equal?

Question

I'm considering edge percolation on $ℤ^2$ with parameter $p$, so that edges are present with probability $p$ and $\theta_x(p)=P[\lvert C(x)\rvert=∞]$ (where $C(x)$ is the component containing $x$ in the percolation graph).

Is it known that $\theta_x(p)=P[\lvert C(x)\rvert=∞]=P[\lvert C(0)\rvert=∞]$ (where 0 is the origin in $ℤ^2$) and therefore $\theta_x(p)=\theta_y(p)$ $∀ x,y∈ℤ^2$. But how can this be explicitly proven/explained?

Answer 1

For all x,y , there exists an automorphism that sends x on y (the translation for example), so that the probablities are equals.

Answer 2

The translation invariance property you are looking for is hard to prove in a single post. I do not know a good source which proves this rigorously for just percolation, however in these lecture notes on the random cluster model/Ising model (which is a more general framework than percolation), translation invariance for infinite volume measures is handled within the first 16 pages.