26 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
Sensitive bootstrap percolation second term
In modified two-neighbour bootstrap percolation in two dimensions each site
of is initially independently infected with probability and
on each discrete time step one additionally infects sites with at least two
non-opposite infected neighbours. In this note we establish that for this model
the second term in the asymptotics of the infection time unexpectedly
scales differently from the classical two-neighbour model, in which arbitrary
two infected neighbours are required. More precisely, we show that for modified
bootstrap percolation with high probability as it holds that for some
positive constant , while the classical model is known to lack the
logarithmic factor.Comment: 9 pages, 1 figur
The maximal running time of hypergraph bootstrap percolation
We show that for every , the maximal running time of the
-bootstrap percolation in the complete -uniform hypergraph on
vertices is . This answers a recent question of Noel
and Ranganathan in the affirmative, and disproves a conjecture of theirs.
Moreover, we show that the prefactor is of the form
as .Comment: 10 pages, 2 figures, improved presentatio
Strong Ramsey Games in Unbounded Time
For two graphs and the strong Ramsey game on the
board and with target is played as follows. Two players alternately
claim edges of . The first player to build a copy of wins. If none of
the players win, the game is declared a draw. A notorious open question of Beck
asks whether the first player has a winning strategy in
in bounded time as . Surprisingly, in a recent paper Hefetz
et al. constructed a -uniform hypergraph for which they proved
that the first player does not have a winning strategy in
in bounded time. They naturally ask
whether the same result holds for graphs. In this paper we make further
progress in decreasing the rank.
In our first result, we construct a graph (in fact )
and prove that the first player does not have a winning strategy in
in bounded time. As an application of this
result we deduce our second result in which we construct a -uniform
hypergraph and prove that the first player does not have a winning
strategy in in bounded time. This improves the
result in the paper above.
An equivalent formulation of our first result is that the game
is a draw. Another reason for interest
on the board is a folklore result that the disjoint
union of two finite positional games both of which are first player wins is
also a first player win. An amusing corollary of our first result is that at
least one of the following two natural statements is false: (1) for every graph
, is a first player win; (2) for every graph
if is a first player win, then
is also a first player win.Comment: 17 pages, 48 figures; improved presentation, particularly in section
The second term for two-neighbour bootstrap percolation in two dimensions
In the -neighbour bootstrap process on a graph , vertices are infected
(in each time step) if they have at least already-infected neighbours.
Motivated by its close connections to models from statistical physics, such as
the Ising model of ferromagnetism, and kinetically constrained spin models of
the liquid-glass transition, the most extensively-studied case is the
two-neighbour bootstrap process on the two-dimensional grid . Around 15
years ago, in a major breakthrough, Holroyd determined the sharp threshold for
percolation in this model, and his bounds were subsequently sharpened further
by Gravner and Holroyd, and by Gravner, Holroyd and Morris.
In this paper we strengthen the lower bound of Gravner, Holroyd and Morris by
proving that the critical probability for percolation
in the two-neighbour model on satisfies The proof of
this result requires a very precise understanding of the typical growth of a
critical droplet, and involves a number of technical innovations. We expect
these to have other applications, for example, to the study of more general
two-dimensional cellular automata, and to the -neighbour process in higher
dimensions.Comment: 53 pages, 6 figures, 1 appendi
Computing the Difficulty of Critical Bootstrap Percolation Models is NP-hard
Bootstrap percolation is a class of cellular automata with random initial state. Two-dimensional bootstrap percolation models have three universality classes, the most studied being the `critical' one. For this class the scaling of the quantity of greatest interest -- the critical probability -- was determined by Bollobás, Duminil-Copin, Morris and Smith in terms of a combinatorial quantity called `difficulty', so the subject seemed closed up to finding sharper results. In this paper we prove that computing the difficulty of a critical model is NP-hard and exhibit an algorithm to determine it, in contrast with the upcoming result of Balister, Bollobás, Morris and Smith on undecidability in higher dimensions. The proof of NP-hardness is achieved by a reduction to the Set Cover problem