# On bernoulli percolation, increasing events and Russo's formula

I am very new to this particular branch of probability theory, I try to be as formal as possible. In this question I consider bernoulli percolation as it is usually introduced as a first model (see for instance Geoffrey Grimmett).

Problem: Let $x,y \in \mathbb{Z}^d$. Prove that $f(p):= \mathbb{P}_p( x \leftrightarrow y)$ is strictly increasing in $p \in[0,1]$.

My approach: First of, let me just state that it is clear that $f(p)$ is increasing in $p$. Both, intuitively and rigorously. The event $x \leftrightarrow y$ (there exists an open path from $x$ to $y$) is an increasing event, i.e. opening up edges is beneficial for the event $x \leftrightarrow y$ and it is a straightforward result from percolation theory that if $A$ is an increasing event, then $p \in [0,1] \mapsto \mathbb{P}_p(A)$ is increasing.

The issue of course being that I want to establish that $\mathbb{P}_p(A)< \mathbb{P}_q(A)$ for $p<q$.

Here I consider it to be a good idea to use Russo's formula:

Theorem (Russo's formula): Let $A \in \mathcal{F}_E$ be an increasing event depending on the edges in a finite subset $F \subset E$ only. Then $p \mapsto \mathbb{P}_p(A)$ is differentiable, and \begin{align} \frac{d}{dp} \mathbb{P}_p(A) = \sum_{e \in F} \mathbb{P}_p(e \text{ is pivotal for }A) \tag{*}\end{align}

If I can use this formula to prove that $f'(p)>0$, then indeed $f$ is strictly increasing. Of course the event $A = \{x \leftrightarrow y \}$ is increasing and depends only on finitely many edges.

Consider dimension $d=2$, then I identified pivotal edges $e \in E$ as follows: An edge $e$ is pivotal (essential) for the event $x \leftrightarrow y$ if and only if there exists an open path from $x$ to $y$ going through the open edge $e$ (say $\gamma$) and there exists a dual open path (say $\gamma^*$) that connects the two endpoints of $e^*$ and is a circuit containing the point $x$.

The picture below (taken from P. Nolin Percolation) is related to said situation, it depicts the event $0 \leftrightarrow \partial B_n$ and shows the paths $\gamma$ and $\gamma^*$.

My Question: How do I complete the proof? I would be happy to understand it even just in the case of $d=2$. So far I have: $$f'(p) = \sum_{e \in F} \mathbb{P}_p( \exists \gamma, \exists \gamma^*) \overset{FKG}\geq\sum_{e \in F} \mathbb{P}_p( \exists \gamma) \mathbb{P}_p( \exists \gamma^*) \overset{?}>0$$