Existence of correlation length in percolation I saw that there exists a constant $\xi = \xi(p)$, the correlation length, such that for the percolation of parameter $p$ on $\mathbb{Z}^d$, if we note $e_1 = (1,0,...,0)$, we have for any $n\geq 1$
$$\mathbb{P}_p [0\leftrightarrow ne_1] \leq e^{-n/\xi(p)}.$$
Unfortunately, I could not find a proof of this result. I know that we have exponentiel decay of the radius of the cluster below $p_c$, but I would like to prove it without using this stronger result.
 A: Define for any $n\geq 1$ the sequence $u_n := \mathbb{P}_p (0\leftrightarrow ne_1)$. Recall that the FKG inequality says that if $A,B$ are two increasing (or decreasing) events, then $$\mathbb{P}(A\cap B)\geq \mathbb{P} (A) \cdot \mathbb{P}(B).$$
Remark now that the sequence $\left(\dfrac{1}{u_n}\right)$ is super-multiplicative in the sense that $u_{n+m} \geq u_n u_m$.
Indeed, assuming that $n < m$, we have
$$u_n u_m = \mathbb{P}(0\leftrightarrow ne_1) \mathbb{P}(0\leftrightarrow me_1)= \mathbb{P}(0\leftrightarrow ne_1)\mathbb{P}(n\leftrightarrow (n+m)e_1).$$
Now, we can use the FKG inequality to obtain
$$u_n u_m \leq \mathbb{P}\left(\lbrace 0\leftrightarrow ne_1\rbrace \cap \lbrace n\leftrightarrow (n+m)e_1\rbrace\right) \leq \mathbb{P}(0\leftrightarrow (n+m)e_1) = u_{n+m}.$$
Using the Fekete's (also known as subadditive) lemma, we know that $u_n ^{1/n}$ converges to $u:= \sup u_n^{1/n}$. Therefore, writing $u = \exp[-\frac{1}{\xi (p)}]$ we get that for all $n\geq 1$,
$$u_n \leq \exp\left[-\frac{n}{\xi (p)}\right],$$
since $u_n \leq u^n$ for any $n\geq 1$.
