Two Phase Transitions in Two-way Bootstrap Percolation

09/26/2018

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by   Ahad N. Zehmakan, et al.

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Consider the d-dimensional torus T_L^d and an initial random configuration, where each node is black with probability p and white otherwise, independently. In discrete-time rounds, each node gets black if it has at least r black neighbors and white otherwise, for some 1≤ r≤ d. This basic process is known as two-way r-bootstrap percolation. We prove the process exhibits a threshold behavior with two phase transitions. More precisely, for p≪ p^(1) the process reaches fully white configuration, p^(1)≪ p≪ p^(2) results in the stable coexistence of both colors, and p^(2)≪ p outputs fully black configuration asymptotically almost surely.