I'm looking for ideas on how to tackle this problem:
I'm working on standard independent bernoulli percolation in $\mathbb{Z}^d$.
Russo's formula states that, if an increasing event $A$ depends only on a finite amount of edges, then $$\frac{d}{dp}P_p(A) = \sum_{e \in E}P_p(e \mbox{ is pivotal to }A) = \mathbb{E}(N(A))$$
where $N(A)$ denotes the number of pivotal edges for the event $A$.
A proof of this result can be found in Grimmett's book "Percolation", 1989. It essentially uses the standard coupling of independent percolation and the chain rule for differentiation.
The Question: Now, suppose $A$ is an increasing event that depends on an infinite amount of edges. If $P_p(A)$ as a function of $p$ is differentiable, does Russo's formula work?
In Grimmett's book we have a partial answer: Fix a set $E$ of finite edges and make $$p_E = \left\{\begin{matrix}p &\mbox{ if }& e \notin E\\ p+\delta &\mbox{ if }& e \in E\end{matrix}\right.$$
Then since $A$ is an increasing event we have $$\frac{1}{\delta}[P_{p+\delta}(A)-P_p(A)]\geq \frac{1}{\delta}[P_{p_E}(A)-P_p(A)] \to \sum_{e \in E}P_p(e \mbox{ is pivotal to }A), \mbox{ as }\delta \to 0.$$ Therefore, making $E \uparrow E_{\mathbb{Z}^d}$ we find that $$\liminf_{\delta \to 0} \frac{1}{\delta}[P_{p+\delta}(A)-P_p(A)]\geq \mathbb{E}[N(A)]$$
Assuming the limit exists we can exchange the $\liminf$ by the limit. Therefore we only need the opposite bound.
Sadly I have no clue on how to proceed. I thought maybe it would be possible to find a function $C(E)$ such that $$\frac{1}{\delta}[P_{p+\delta}(A)-P_p(A)]\leq \frac{C(E)}{\delta}[P_{p_E}(A)-P_p(A)]$$
with $C(E) \to 1$ as $E \uparrow E_{\mathbb{Z}^d}$ uniformly on $\delta$. But I have no clue on how to do it (and I'm not convinced it is even possible, as the exchange between $P_{p+\delta}$ and $P_{p_E}$ changes the edges in the infinite set $E_{\mathbb{Z}^d}\setminus E$).
Unfortunately, we can't do better than that, because the probability of an increasing event depending on infinitely many edges needs not be differentiable, such as the probability of an infinite cluster at 0, at p_c.