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On the critical value function in the divide and color model

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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Submitted on : Thursday, January 25, 2018 - 6:48:01 PM
Last modification on : Thursday, June 18, 2020 - 10:18:03 AM
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  • HAL Id : ensl-00624097, version 2
  • ARXIV : 1109.3403



András Bálint, Vincent Beffara, Vincent Tassion. On the critical value function in the divide and color model. ALEA : Latin American Journal of Probability and Mathematical Statistics, Instituto Nacional de Matemática Pura e Aplicada, 2013, 10 (2), pp.653-666. ⟨ensl-00624097v2⟩



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