These fascinating questions have been studied recently, e.g. by Bauerschmidt, Crawford, Taylor and Swan (no percolation on $\mathbb{Z}^2$) and by Bauerschmidt, Crawford and Taylor (percolation phase transition on $\mathbb{Z}^d$ for $d\geq 3$).

These fascinating questions have been studied recently, e.g. by Bauerschmidt, Crawford, Helmuth and Swan (no percolation on $\mathbb{Z}^2$) and by Bauerschmidt, Crawford and Helmuth (percolation phase transition on $\mathbb{Z}^d$ for $d\geq 3$).