Corner percolation with preferential directions

Régine Marchand (1) , Irène Marcovici (1) , Pierrick Siest (1)

1 IECL - Institut Élie Cartan de Lorraine

Corner percolation is a dependent bond percolation model on Z^2 introduced by Bálint Tóth, in which each vertex has exactly two incident edges, perpendicular to each other. Gábor Pete has proven in 2008 that under the maximal entropy probability measure, all connected components are finite cycles almost surely. We consider here a regime where West and North directions are preferred with probability p and q respectively, with (p, q) = (1/2 , 1/2). We prove that there exists almost surely an infinite number of infinite connected components, which are in fact infinite paths. Furthermore, they all have the same asymptotic slope (2q−1)/(1−2p).

Mots clés: constrained percolation; corner percolation

Domaines: Probabilités [math.PR]

Régine Marchand, Irène Marcovici, Pierrick Siest. Corner percolation with preferential directions. 2022. ⟨hal-03890722⟩