Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Tuesday, May 15, 2018, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Ahad N. Zehmakan

Bootstrap Percolation

Given a graph G and an initial configuration where each node is active or inactive, in the r-bootstrap percolation process in each round an inactive node with at least 𝑟 active neighbors becomes active and stays active forever. In the reversible r-bootstrap percolation, each node gets active if it has at least r active neighbors, and inactive otherwise. We consider the following question on the d-dimensional torus: what is the minimum number of nodes which must be initially active to make the whole graph eventually active? We provide asymptotically tight lower and upper bounds. In particular our results settle an open problem by Balister, Bollobas, Johnson, and Walters and generalize the results by Flocchini et al. Then, we consider the random setting where initially each node is active independently with probability p and inactive otherwise. What is the minimum value of p for which with high probability the torus eventually becomes fully active? To address this question, we discuss that the process exhibits a threshold behavior with two phase transitions. This answers an old open problem by Schonmann.

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