493 research outputs found
Finite-size effects for anisotropic bootstrap percolation: logarithmic corrections
In this note we analyze an anisotropic, two-dimensional bootstrap percolation
model introduced by Gravner and Griffeath. We present upper and lower bounds on
the finite-size effects. We discuss the similarities with the semi-oriented
model introduced by Duarte.Comment: Key words: Bootstrap percolation, anisotropy, finite-size effect
The time of graph bootstrap percolation
Graph bootstrap percolation, introduced by Bollob\'as in 1968, is a cellular
automaton defined as follows. Given a "small" graph and a "large" graph , in consecutive steps we obtain from by
adding to it all new edges such that contains a new copy of
. We say that percolates if for some , we have .
For , the question about the size of the smallest percolating graphs
was independently answered by Alon, Frankl and Kalai in the 1980's. Recently,
Balogh, Bollob\'as and Morris considered graph bootstrap percolation for and studied the critical probability , for the event that
the graph percolates with high probability. In this paper, using the same
setup, we determine, up to a logarithmic factor, the critical probability for
percolation by time for all .Comment: 18 pages, 3 figure
Strong-majority bootstrap percolation on regular graphs with low dissemination threshold
International audienceConsider the following model of strong-majority bootstrap percolation on a graph. Let r ≥ 1 be some integer, and p ∈ [0, 1]. Initially, every vertex is active with probability p, independently from all other vertices. Then, at every step of the process, each vertex v of degree deg(v) becomes active if at least (deg(v) + r)/2 of its neighbours are active. Given any arbitrarily small p > 0 and any integer r, we construct a family of d = d(p, r)-regular graphs such that with high probability all vertices become active in the end. In particular, the case r = 1 answers a question and disproves a conjecture of Rapaport, Suchan, Todinca and Verstraëte [38]
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