Threshold behavior of bootstrap percolation

Ahad N. Zehmakan

Abstract: Consider a graph G<math><mi is="true">G</mi></math> and an initial random configuration, where each node is black with probability p<math><mi is="true">p</mi></math> and white otherwise, independently. In discrete-time rounds, each node becomes black if it has at least r<math><mi is="true">r</mi></math> black neighbors and white otherwise. We prove that this basic process exhibits a threshold behavior with two phase transitions when the underlying graph is a d<math><mi is="true">d</mi></math>-dimensional torus and identify the threshold values.