On the location of the infinite cluster in independent percolation My question is about whether something is feasible or not, so I'll give a little bit of context
The problem I'm concerned with is independent percolation in $\mathbb{Z}^d$: which means the following:

(Ignore this part if you already know what percolation is)
Consider the graph whose vertex are $\mathbb{Z}^d$ and there's an edge between the vertex $x$ and $y$ whenever $\sum_{i=1}^d |x_i-y_i|=1$. This model is called "nearest neighbours" percolation.
Given a fixed parameter $p\in [0,1]$, we will assign to every edge $e$, independently from the other edges, a random variable $\omega(e) \in \{0,1\}$ such that
$$p = P_p(\omega(e) = 1) = 1-P_p(\omega(e)=0).$$
Now, we are interested in the random subgraph formed by taking the graph formed by discarding all edges in which $\omega(e)=0$. In particular we want to know whether there's an infinite cluster made out of "open edges" (edges where $\omega(e)=1$) or not. When there's an infinite cluster we say that the model "percolates" and we have a special notation for the probability of this event:
$$\theta(p) = P_p(0 \leftrightarrow \infty) = P_p(|C(0)|=\infty).$$
(here $C(0)$ denotes the set of vertices connected to $0$ by open edges)
Now, it is easy to show that, for $d\geq 2$, if we take $p$ close to $0$ there's almost surelly no infinite cluster, and if $p$ is close enough to $1$ there's positive probability that the origin is inside an infinite cluster. Burton and Keane (1989) showed that, if $\theta(p)>0$ then there exists exactly one  infinite cluster. The argument uses ergodicity and the property that this graph is amenable (the boundary of a box increases in a slower rate than it's interior).

Enough context, now comes my problem: We know that if $\theta(p)>0$ then there exists an infinite cluster almost surely. This implies that if we take a box $B(n) = [-n,n]^d$ and look at the event where some vertex of this box lies in the infinite cluster $\{B(n) \leftrightarrow \infty\}$, then we have that
$$\lim_{n \to \infty}P_p(B(n)\leftrightarrow \infty) = P_p\left(\bigcup_{n=1}^\infty \{B(n)\leftrightarrow \infty\}\right) = P_p(\exists \mbox{ infinite cluster}) = 1$$
so, if $\theta(p)>0$, there exists an $m = m(\varepsilon, p)$ such that $P_p(B(m)\leftrightarrow \infty)>1-\varepsilon$.

The issue is: Is there a way to calculate who this $m$ is?

My doctoral advisor believes that it should be possible to approximate anything that doesn't use the axiom of choice, so it should be possible to estimate this $m$ in some way. But the thing is that the question is about the location of the cluster, and everything in the literature seems to be concerned only on the existence of such cluster.
The argument to show that there exists one cluster essentially is: We are in a product space, invariant under translations. Our model is mixing, hence ergodic and as the number of infinite clusters is invariant under translations it is constant almost surely. Then some simple arguments eliminate the possibility that there's $2\leq k<\infty$ infinite clusters, and the ameable property is used to exclude an infinite number of infinite clusters. In no point there's information about the location of the cluster.
I have absolutelly no idea on how to approach this. Does anyone have ideas on the matter?

One way to think about this is that, by Birkhoff's Ergodic Theorem, if we consider
$$K(n) = |\{x \in B(n): x \leftrightarrow \infty\}|$$
then
$$\frac{K(n)}{|B(n)|}\to \theta(p) a.s.$$
and so, if we know how the convergence in Birkhoff's ergodic theorem goes, we can attempt to find $n$ such that
$$P(K(n)\geq 1)>1-\varepsilon.$$
but again, no clue on how to do that.
Thanks in advance.
 A: Let us discuss the special case $d=2$ first. Here, the phase picture is clear: We have $\theta(p)>0$ iff $p>0.5$, and in this case the dual model on the dual lattice of $\mathbb{Z}^2$ is in the subcritical phase, meaning that for dual vertices $v^*,w^*$ we have that
$$P_p[v^*\leftrightarrow w^*] \leq C \exp(-c|v^*-w^*|)$$
for some constants $C,c>0$ that only depend on $p$. This property is called exponential decay of correlations and the fact that the dual model satisfies this property is widely known, see any introductory text on percolation or the random cluster model (in the plane).
Now, the event
$$\{B(n) \not\leftrightarrow \infty\}$$
is equal to
$$\{B(n) \text{ is engulfed by a dual cluster}\}$$
and it is easy to show that the probability for this event also decays exponentially fast w.r.t. $n$, which gives you a pretty good handle regarding $m=m(\epsilon,p)$.

For $d\geq 3$, I have to admit that I am not an expert on the topic. I do believe that a reasonable bound on $m$ should be obtainable, though I highly doubt that it could be possible with just using the information $\theta(p) > 0$. For instance, your calculation
$$ \lim_{n\to\infty} P_p(B(n)) = \ldots = 1$$
seems sketchy to me, or uses arguments I am not aware of. It should be easy to construct a measure on the graph $\mathbb{Z}^d$ that has $\theta(p) >0 $, but $P_p(B(n))$ stays small for increasing $n$. Here I am thinking of a measure that produces a connected infinite component with "holes" with diameter at every length scale (scaling invariant holes, so to say), such that the probability of $B(n)$ being in such a hole is not decaying to zero with increasing $n$.
In your situation, I would suggest to look for a rigorous proof of the claim $P_p(B(n))\to 1$ because to me it looks like there has to be more to it than just $\theta(p) >0$.
