While I was reading Introduction to Bernoulli Percolation by Duminil-Copin (https://www.ihes.fr/~duminil/publi/2017percolation.pdf), I got stuck with the proof of the following part of the proof of Theorem 1.1,which implies that $p_c(d)<p_c(2)$, for $d \geq 2$:
The argument showing that $p_c < 1$ is more elaborated. We need to prove that when $p$ is close to 1, then $θ(p) > 0$. Any percolation on $\mathbb{Z}^ d$ contains a copy of Bernoulli percolation on $\mathbb{Z}^ 2$ (simply look at the restriction on $ω ∩ \mathbb{Z}^ 2$ ). Hence, if the probability of $0 ←→ ∞ $ on $\mathbb{Z}^ 2$ is strictly positive, then $θ(p) > 0$ on $\mathbb{Z}^ d$ , and therefore $ p_c(d) ≤ p_c(2)$. It is sufficient to show the result for $d = 2$.
Howover, it seems to me that even an infinite cluster on $\mathbb{Z}^2$ is equivalent to one on $(\mathbb{Z}^2,0)$, the equation $\mathbb{P}(|C| = +\infty \text{ and $C$ is on $\mathbb{Z}^2$}) =\mathbb{P}(|C| = +\infty \text{ and $C$ is on $(\mathbb{Z}^2,0,0,...,0)$}) $ is not very obvious as the latter one is actually an event in a "larger" universe (having higher dimensions) so that their probability may not be necessary the same.
I also have checked some other textbooks such the one by Grimmett, though they all have used a similar method to prove this theorem, none of them has given a more rigorous proof for this part.
Can anyone give me a better proof or tell me the right method if my idea is wrong?
Let $(X,F,\mu)$ and $(Y,G,\nu)$ be probability spaces, and denote by $\lambda=\mu\otimes\nu$ the product measure on $X \times Y$. Then for every event $A \in F$, we have $\lambda(A \times Y)=\mu(A)$.
That is all which is used here, since the edge variables are independent.