Does Eulerian percolation on Z 2 percolate

Eulerian percolation on Z(2) with parameter p is the Bernoulli bond percolation with parameter p conditioned on the event that every site has an even degree. Eulerian percolation with parameter p coincides with the contours of the Ising model for a well-chosen parameter beta(p). This allows to study the percolation properties of Eulerian percolation in the ferromagnetic range beta(p) >= 0, corresponding to p <= 1/2. To study the case p > 1/2, we provide a new coupling between Bernoulli percolations with parameters p and 1 - p, that increases connectivity properties and preserves the parity of the degrees of the sites. Some key ingredients of the proofs are couplings between Eulerian percolation, the Ising model and FK-percolation.