frank kindle
Let $\mathbb{L}^d=(\mathbb{Z}^d, E)$ be a graph where an edge ${x,y}$ is in $E$ if $d(x,y)=1$. Let $(\omega e){e\in E}$ be i.i.d. Bernoulli random variables of parameter $p$. We say an edge $e$ is open if $\omega _e =1$. For a given $x$ in $\mathbb{Z}^d$, let $C_x$ be the set of vertices that are connected to $x$ via a path that has only open edges. Let $\theta (p)$ be defined as $\mathbb{P} (|C_0|=\infty)$.\
My question is: how do we know that $|C_x|$ is a random variable for each $x$? And how do we know that the probability on that is well defined? That is, let there be two different probability spaces $A_1$ and $A_2$, both satisfying everything that is said above. How can I know that, $\mathbb{P}_1 (|C_0|=\infty) = \mathbb{P}_2 (|C_0|=\infty)$ (for a given $p$)?\
The reason behind this question is that there is a proof that relies on adding structure to percolation. It begins by defining $U_e$ uniform iid random variables, such that $\omega_e = 1$ iff $U_e < p$. Those weren't there at the beggining so it is not obvious that this new object is actually the same percolation. The same can be said for many other proofs, that basically study one particular case of percolation model (with more structure) and then claims that the result is also true for all percolation models.
placid imp
quaint charm
maiden river
magic olive
