= Solution
The event that $T_1$ is connected is translation invariant and hence has probability zero or one. Layer-exchange symmetry gives the same probability for connectivity of $T_2$. If $T_1$ were connected almost surely, both induced forests would therefore be connected almost surely.
On that event the spanning tree must contain exactly one vertical edge: it needs at least one to join the two layers, while two vertical edges together with the unique paths inside the connected $T_1$ and $T_2$ would form a cycle. But a translation-invariant random set cannot contain exactly one vertical edge almost surely. Every specified vertical edge would have probability zero of being the unique one, and their countable union would still have probability zero. This contradiction proves that $T_1$ is almost surely not connected.
Back to article page
