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On the critical value function in the divide and color model

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Abstract

The 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.
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Dates and versions

ensl-00624097 , version 1 (15-09-2011)
ensl-00624097 , version 2 (25-01-2018)

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András Bálint, Vincent Beffara, Vincent Tassion. On the critical value function in the divide and color model. 2011. ⟨ensl-00624097v1⟩
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