These fascinating questions have been studied recently, e.g. by [Bauerschmidt, Crawford, Taylor and Swan][1] (no percolation on $\mathbb{Z}^2$) and by [Bauerschmidt, Crawford and Taylor][2] (percolation phase transition on $\mathbb{Z}^d$ for $d\geq 3$). 


  [1]: https://arxiv.org/abs/1912.04854
  [2]: https://arxiv.org/abs/2107.01878