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At my probability theory proseminar I had a speech about the percolation theory and one of the topics I presented was the uniqueness of an infinite open cluster in Bernoulli bond percolation in $\mathbb{Z}^d$. However, I had real problems with proving that N (the number of infinite open clusters) is constant a.s. I tried using Kolmogorov's 0-1 law, but it only tells us that $\mathbb{P}_p(N \geq 1) \in \{0, 1\}$.

My speech was mainly based on Grimmett's Probability on Graphs (http://www.statslab.cam.ac.uk/~grg/books/USpgs-rev3.pdf, theorem 5.22). The proof says that $\{N = c\}$ is a translation-invariant event, therefore $\mathbb{P}(N = c) \in \{0, 1\}$ for any $c$, but I don't understand why is the sample space $\Omega$ in a product form. In fact, the codomain of a stochastic process representing the percolation model is $\{0, 1\}^{\mathbb{E}^d}$ and this is in a product form, $\textbf{not}$ the $\Omega$ itself. This makes me even more confused, since I'm a complete greenhorn in ergodic theory. I will highly appreciate any kind of explanation or reference to some more elaborate work in this context.

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You use the fact that the product measure on $\{0,1\}^{\mathbb{E}^d}$ is ergodic with respect to the shifts.

The ergodicity of the product measure w.r.t. the shifts means (by definition) that $\mathbb{P}_p(E)\in\{0,1\}$ for any shift-invariant event $E$. To show that the product measure is ergodic, you approximate $E$ with a cylinder set $C$. Now, if you shift $C$ sufficiently far, you get another cylinder set that is independent of $C$. This will show that the approximated set $E$ (which is shift-invariant) is independent of itself.

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p. 39 states that the sample space $\Omega$ is $\{0, 1\}^{E^d}$ so I would say it is by definition.

You can understand why the event $\{N = c\}$ is translation invariant as follows. Take any realization of the sample space that has $c$ clusters. Then if you translate that realization you will find another realization still in the set. In other words, translation is a bijection on $\{N = c\}$.

To prove that set has measure $0$ or $1$ you will need to use the Kolmogorov $0-1$ law. See this thread for suggestion on how to do it.

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  • $\begingroup$ Okay, I think that now I understand the notation - we define from scratch the sample space to have $\Omega = \{0, 1\}^{\mathbb{E}^d}$ and the appropriate $\sigma$-algebra $\mathcal{F}$ of cylinder sets and a probability measure $\mathbb{P_p}$ on $\mathcal{F}$. How can I obtain the constantness of $N$? The random variable $N$ is invariant under the translations of $\{0, 1\}^{\mathbb{E}^d}$, but I haven't any knowledge about the ergodic theorems. $\endgroup$ – tosi3k Jun 4 '15 at 20:06
  • $\begingroup$ @tosi3k I updated to explain that as well. $\endgroup$ – muaddib Jun 4 '15 at 20:21
  • $\begingroup$ I see why the events $\{N = c\}$ are translation invariant, but I have no knowledge why it implies $\mathbb{P}(N = c) \in \{0, 1\}$. $\endgroup$ – tosi3k Jun 4 '15 at 20:50
  • $\begingroup$ @tosi3k Updated. $\endgroup$ – muaddib Jun 4 '15 at 21:40

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