Does Eulerian percolation on Z² percolate?
Résumé
Eulerian percolation on Z2 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 β(p). This allows to study the percolation properties of Eulerian percolation in the ferromagnetic range β(p) ≥0, correspond- ing 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.