Uniqueness of the Stationary Point

Let $G$ be a finite connected graph without bridges, and let $a,b$ be two non-adjacent vertices. Consider Bernoulli bond percolation on $G$, where each edge is open independently with probability $p \in [0,1]$. Let $f(p)$ denote the probability that $a$ is connected to $b$ by an open path.
The function $f(p)$ is a polynomial in $p$, and under the above assumptions one has $f'(0)=f'(1)=0$. Therefore, the equation
$$ f'(p)=p $$
has at least one solution in $(0,1)$. The main result states that this solution is unique.
The talk will present an outline of the proof based on (currently unpublished) work by A. Teixeira et al. The argument is highly non-elementary, and no elementary proof is known.