Let $x \in \lbrack 0,1 \rbrack$. Then for any finite graph $G$ consider the ErdösRenyi random graph where we independently keep each of the edges with probability $x$. Denote the corresponding probability measure $P_x$. An even spanning subgraph of $G$ is a subgraph where all the vertices are kept and some of the edges such that each vertex has even degree. Denote the set of even spanning subgraphs of G by $\text{ess}(G)$. For an even spanning subgraph $g$ let $O(g)$ be the event that all the edges are kept. By some knowledge about geometric representations of the Ising model I happen to know that for every pair of vertices $v_1,v_2 \in G$ we have \begin{align*} \sum_{g_1, g_2 \in \text{ess}(G)} \frac{ P_{x^2} \left( \{v_1 \leftrightarrow v_2 \}, O(g_1), O(g_2) \right)}{P_x(O(g_1))} = \sum_{g_1, g_2 \in \text{ess}(G)} P_x( \{v_1 \leftrightarrow v_2 \}, O(g_1)) P_x( \{v_1 \leftrightarrow v_2 \}, O(g_2)) \end{align*} Here $\{v_1 \leftrightarrow v_2 \}$ is the event that there is a path of kept edges in $G$ connecting $v_1$ and $v_2$. However, this is a problem purely on ErdösRenyi random graphs, but I can't find a combinatorial proof of this which could maybe give some insights on how the relation generalizes beyond the class of events $\{ v_1 \leftrightarrow v_2 \}$.

$\begingroup$ The bracket in the denominator is not closed, and in the numerator too. Also am I correct that the event in the numerator read as "the edge $\{ v_1 \leftrightarrow v_2 \}$ is kept and either all edges of $g_1$ are kept or all edges of $g_2$ are kept or both"? $\endgroup$– Fedor PetrovCommented Dec 29, 2019 at 20:24

$\begingroup$ Thanks  I have added an explaination. $\endgroup$– Frederik Ravn KlausenCommented Jan 1, 2020 at 17:34

$\begingroup$ Ah, thank you, then I proved something different, but possibly related. Should think. $\endgroup$– Fedor PetrovCommented Jan 1, 2020 at 19:00

$\begingroup$ Yes  I start to thnik that one should somehow invoke combinatorics like in the switching lemma. $\endgroup$– Frederik Ravn KlausenCommented Apr 4, 2020 at 9:22
1 Answer
(warning: below $\{v_1 \leftrightarrow v_2 \}$ has different meaning than in OP)
Let me prove a slightly different identity
\begin{align*} x\sum_{g_1, g_2 \in \text{ess}(G)} \frac{ P_{x^2} \left( \{v_1 \leftrightarrow v_2 \}, O(g_1), O(g_2) \right)}{P_x(\{v_1 \leftrightarrow v_2 \},O(g_1))} = \\\sum_{g_1, g_2 \in \text{ess}(G)} P_x( \{v_1 \leftrightarrow v_2 \}, O(g_1)) P_x( \{v_1 \leftrightarrow v_2 \}, O(g_2)). \end{align*}
If $E_i$ denotes edge set of $g_i$ and $e=\{ v_1 \leftrightarrow v_2 \}$, then LHS may be written as $$\sum_{g_1,g_2} x^{2E_1\cup E_2\cup eE_1\cup e+1}=\sum_{g_1,g_2} x^{E_2\cup e+(E_1\Delta E_2)\cup e},$$ where we apply the general identity $2A\cup BA=B+A\Delta B$ for $A=E_1\cup e$, $B=E_2\cup e$, and $(E_1\Delta E_2)\cup e=1+A\Delta B$. It remains to change the pair $(g_1,g_2)$ in RHS of my identity onto pairs $(g_1\Delta g_2,g_2)$.
I am still not sure what exactly do you mean (there are several typos at least), but possibly something provable along these lines.