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Consider a graph $G$ and an initial random configuration, where each node isblack with probability $p$ and white otherwise, independently. In discrete-timerounds, each node becomes black if it has at least $r$ black neighbors andwhite otherwise. We prove that this basic process exhibits a threshold behaviorwith two phase transitions when the underlying graph is a $d$-dimensional torusand identify the threshold values.

Two Phase Transitions in Two-way Bootstrap Percolation