103 research outputs found
Maximal Bootstrap Percolation Time on the Hypercube via Generalised Snake-in-the-Box
In -neighbour bootstrap percolation, vertices (sites) of a graph are
infected, round-by-round, if they have neighbours already infected. Once
infected, they remain infected. An initial set of infected sites is said to
percolate if every site is eventually infected. We determine the maximal
percolation time for -neighbour bootstrap percolation on the hypercube for
all as the dimension goes to infinity up to a logarithmic
factor. Surprisingly, it turns out to be , which is in great
contrast with the value for , which is quadratic in , as established by
Przykucki. Furthermore, we discover a link between this problem and a
generalisation of the well-known Snake-in-the-Box problem.Comment: 14 pages, 1 figure, submitte
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
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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
Cooperative Behavior of Kinetically Constrained Lattice Gas Models of Glassy Dynamics
Kinetically constrained lattice models of glasses introduced by Kob and
Andersen (KA) are analyzed. It is proved that only two behaviors are possible
on hypercubic lattices: either ergodicity at all densities or trivial
non-ergodicity, depending on the constraint parameter and the dimensionality.
But in the ergodic cases, the dynamics is shown to be intrinsically cooperative
at high densities giving rise to glassy dynamics as observed in simulations.
The cooperativity is characterized by two length scales whose behavior controls
finite-size effects: these are essential for interpreting simulations. In
contrast to hypercubic lattices, on Bethe lattices KA models undergo a
dynamical (jamming) phase transition at a critical density: this is
characterized by diverging time and length scales and a discontinuous jump in
the long-time limit of the density autocorrelation function. By analyzing
generalized Bethe lattices (with loops) that interpolate between hypercubic
lattices and standard Bethe lattices, the crossover between the dynamical
transition that exists on these lattices and its absence in the hypercubic
lattice limit is explored. Contact with earlier results are made via analysis
of the related Fredrickson-Andersen models, followed by brief discussions of
universality, of other approaches to glass transitions, and of some issues
relevant for experiments.Comment: 59 page
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