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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 $\gamma^*$.

related picture

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: $$ f'(p) = \sum_{e \in F} \mathbb{P}_p( \exists \gamma, \exists \gamma^*) \overset{FKG}\geq\sum_{e \in F} \mathbb{P}_p( \exists \gamma) \mathbb{P}_p( \exists \gamma^*) \overset{?}>0 $$

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I think it's better if you use the 'standard coupling' of percolation and express the event (non strictly increasing) in terms of the uniforms only to realize the proba is 0. (the proof wont depend on dimension, and will be valid on many graph). Also, the events you have considered are not both increasing (one is the other is decreasing). so the FKG inequality will not provide you with the bound you want.

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