We're in $\mathbb{Z}^d$, and we have a collection $(X_e^p)_{e \in E}$ where $P(X_e^p=1)=p=1-P(X_e^p=0)$, i.e. the probability of an edge $e$ being open is $p$.
Let $C^p(y)$ is a random open cluster, i.e. the set of all points in $\mathbb{Z}^d$ which are connected to $y \in \mathbb{Z}^d$ through open edges.
Why do we have $P(\{\# C^p(y)=\infty\})=P(\{\# C^p(0)=\infty\})$?
The book I'm reading states that it's due to translation invariance of the lattice... but I have no idea what that is, why that happens, and my web searches came up empty.
Any help would be appreciated.
Edit: I've edited this question since I have another doubt, highly related to this one, and there would be no point of repeating the description of the context.
The book defines $C^p(x)=\{y \in \mathbb{Z}^d: x\leftrightarrow_p y\}$, where $y\leftrightarrow_p x$ means there is a open path between $y$ and $x$. I think I understand this, but then the author proceeds to state $\#C^p(x)=\sum_{y \in \mathbb{Z}^d} \mathbf{1}_{\{x\leftrightarrow_p y\}}$, which I'm not sure how to interpret, since $x$ is given, and $y$ is the index of the sum. What is the function $\mathbf{1}_{\{x\leftrightarrow_p y\}}$?
The author doesn't even define the event space, the $\sigma$-algebra, nor define a measure in the generating set of the sigma algebra.