17 research outputs found
Lower bounds for bootstrap percolation on Galton-Watson trees
Bootstrap percolation is a cellular automaton modelling the spread of an
`infection' on a graph. In this note, we prove a family of lower bounds on the
critical probability for -neighbour bootstrap percolation on Galton--Watson
trees in terms of moments of the offspring distributions. With this result we
confirm a conjecture of Bollob\'as, Gunderson, Holmgren, Janson and Przykucki.
We also show that these bounds are best possible up to positive constants not
depending on the offspring distribution.Comment: 7 page
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
Algebraic and combinatorial expansion in random simplicial complexes
In this paper we consider the expansion properties and the spectrum of the
combinatorial Laplace operator of a -dimensional Linial-Meshulam random
simplicial complex, above the cohomological connectivity threshold. We consider
the spectral gap of the Laplace operator and the Cheeger constant as this was
introduced by Parzanchevski, Rosenthal and Tessler ( 36, 2016).
We show that with high probability the spectral gap of the random simplicial
complex as well as the Cheeger constant are both concentrated around the
minimum co-degree of among all -faces. Furthermore, we consider a
generalisation of a random walk on such a complex and show that the associated
conductance is with high probability bounded away from 0.Comment: 28 page
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Extremal and probabilistic bootstrap percolation
In this dissertation we consider several extremal and probabilistic problems in bootstrap percolation on various families of graphs, including grids, hypercubes and trees. Bootstrap percolation is one of the simplest cellular automata. The most widely studied model is the so-called r-neighbour bootstrap percolation, in which we consider the spread of infection on a graph G according to the following deterministic rule: infected vertices of G remain infected forever and in successive rounds healthy vertices with at least r already infected neighbours become infected. Percolation is said to occur if eventually every vertex is infected.
In Chapter 1 we consider a particular extremal problem in 2-neighbour bootstrap percolation on the n \times n square grid. We show that the maximum time an infection process started from an initially infected set of size n can take to infect the entire vertex set is equal to the integer nearest to (5n^2-2n)/8. In Chapter 2 we relax the condition on the size of the initially infected sets and show that the maximum time for sets of arbitrary size is 13n^2/18+O(n).
In Chapter 3 we consider a similar problem, namely the maximum percolation time for 2-neighbour bootstrap percolation on the hypercube. We give an exact answer to this question showing that this time is \lfloor n^2/3 \rfloor.
In Chapter 4 we consider the following probabilistic problem in bootstrap percolation: let T be an infinite tree with branching number \br(T) = b. Initially, infect every vertex of T independently with probability p > 0. Given r, define the critical probability, p_c(T,r), to be the value of p at which percolation becomes likely to occur. Answering a problem posed by Balogh, Peres and Pete, we show that if b \geq r then the value of b itself does not yield any non-trivial lower bound on p_c(T,r). In other words, for any \varepsilon > 0 there exists a tree T with branching number \br(T) = b and critical probability p_c(T,r) < \varepsilon.
However, in Chapter 5 we prove that this is false if we limit ourselves to the well-studied family of Galton--Watson trees. We show that for every r \geq 2 there exists a constant c_r>0 such that if T is a Galton--Watson tree with branching number \br(T) = b \geq r then
We also show that this bound is sharp up to a factor of O(b) by describing an explicit family of Galton--Watson trees with critical probability bounded from above by C_r e^{-\frac{b}{r-1}} for some constant C_r>0