Given a (first finite and later infinite) graph $G =(V,E)$ the uniform even graph is the uniform probability measure on the set of spanning even subgraphs. That is subgraphs (V, E') with $E' \subset E$ such that every vertex has even degree.
The subgraph $(V, \emptyset)$ is always an even subgraph so this set is non-empty.
Similarly, and odd subgraph is a (spanning) subgraph where all vertices have odd degree. If $G =(V,E)$ has an odd subgraph then we can define the uniform odd subgraph as the uniform probability measure on the set of odd subgraphs.
Notice that if we have one odd subgraph $(V, E_{odd})$ a way to sample the uniform odd subgraph is to sample a uniform even subgraph $\eta$ of $G$ and then taking the symmetric difference $ E_{odd} \triangle \eta$.
We can extend these two notions to infinite graphs, either by taking weak limits as is standard in statistical mechanics or by the following construction of infinite volume uniform even graph:
Notice that the set of all (spanning) subgraphs of a finite or infinite graph $G = (V,E)$ is a group with symmetric difference as the group operation. The set of even graphs is a subgroup of the group. The Haar probability measure on the (compact) subgroup is the uniform even graph. This way we can construct the uniform even subgraph of an infinite graph for example the hypercubic lattice $\mathbb{Z}^d$.
The uniform odd subgraph of $\mathbb{Z}^d$ we can construct by taking the symmetric difference of the uniform even subgraph with a fixed dimer configuration.
We are interested in the percolation of the random infinite graph which is defined as the existence of an infinite component. Equivalently (it turns out), we can ask whether there is an infinite path through 0 with positive probability. In that case we say that the measure percolates.
For $d \geq 3$ we know that both the uniform even subgraph and the uniform odd subgraph percolates (since the marginal of the Haar measure is the Haar measure, see https://arxiv.org/abs/2306.05130).
For $d=2$ since the uniform even subgraph is the interface of an infinite temperature Ising model we can prove with a huge detour that the uniform even subgraph percolates.
For the uniform odd subgraph in two dimensions, we do not know. But we suspect that it does not percolate. See the stark constrast in the pictures below (where the largest component in a sample of the uniform odd and even graphs of a 100x100 grid is colored red)
I therefore ask:
Does the uniform odd subgraph of $\mathbb{Z}^2$ percolate?
Is there an easy self contained proof that the uniform even subgraph of $\mathbb{Z}^2$ percolates?
One approach is to artifically change the boundary conditions to get a line from the middle left to the middle right and prove that the fluctuations of this line are not too large (see The fluctuations of a random path).
