Suppose $T$ is a uniform spanning tree on the graph $G=\mathbb{Z}^2 \oplus \mathbb{Z}^2$, and $T_1$ and $T_2$ are the induced subgraphs of $T$ on the two copies of $\mathbb{Z}^2$. I am trying to show that $T_1$ is almost surely not connected and almost surely contains a component of size at least $N$ for some fixed $N>0$.
These facts both feel intuitively true (as I understand that the probability of a component of size $n$ in $T_1$ goes to zero as $n\to\infty$ but is positive for all $n$) to me but I don't have the first clue how I would go about proving them.