https://hal-ens-lyon.archives-ouvertes.fr/ensl-00624097v2Bálint, AndrásAndrásBálintDepartment of Mathematical Sciences - Chalmers University of Technology [Göteborg] - GU - University of Gothenburg Beffara, VincentVincentBeffaraUMPA-ENSL - Unité de Mathématiques Pures et Appliquées - ENS Lyon - École normale supérieure - Lyon - CNRS - Centre National de la Recherche ScientifiqueTassion, VincentVincentTassionUMPA-ENSL - Unité de Mathématiques Pures et Appliquées - ENS Lyon - École normale supérieure - Lyon - CNRS - Centre National de la Recherche ScientifiqueOn the critical value function in the divide and color modelHAL CCSD2013[MATH.MATH-PR] Mathematics [math]/Probability [math.PR]Beffara, Vincent2018-01-25 18:48:012020-06-18 10:18:032018-03-22 08:48:18enJournal articleshttps://hal-ens-lyon.archives-ouvertes.fr/ensl-00624097v2/documenthttps://hal-ens-lyon.archives-ouvertes.fr/ensl-00624097v1application/pdf2The divide and color model on a graph G arises by first deleting each edge of G with probability (1-p) independently of each other, then coloring the resulting connected components (i.e., every vertex in the component) black or white with respective probabilities r and (1-r), independently for different components. Viewing it as a (dependent) site percolation model, one can define the critical point r_c(p). In this paper, we first give upper and lower bounds for r_c(p) for general G via a stochastic comparison with Bernoulli percolation, and discuss (non-)monotonicity and (non-)continuity properties of r_c(p) in p. Then we focus on the case G=Z^2 and prove continuity of r_c(p) as a function of p in the interval [0,1/2), and we examine the asymptotic behavior of the critical value function as p tends to its critical value.