68 research outputs found
Nucleation scaling in jigsaw percolation
Jigsaw percolation is a nonlocal process that iteratively merges connected
clusters in a deterministic "puzzle graph" by using connectivity properties of
a random "people graph" on the same set of vertices. We presume the
Erdos--Renyi people graph with edge probability p and investigate the
probability that the puzzle is solved, that is, that the process eventually
produces a single cluster. In some generality, for puzzle graphs with N
vertices of degrees about D (in the appropriate sense), this probability is
close to 1 or small depending on whether pD(log N) is large or small. The one
dimensional ring and two dimensional torus puzzles are studied in more detail
and in many cases the exact scaling of the critical probability is obtained.
The paper settles several conjectures posed by Brummitt, Chatterjee, Dey, and
Sivakoff who introduced this model.Comment: 39 pages, 3 figures. Moved main results to the introduction and
improved exposition of section
Random growth models with polygonal shapes
We consider discrete-time random perturbations of monotone cellular automata
(CA) in two dimensions. Under general conditions, we prove the existence of
half-space velocities, and then establish the validity of the Wulff
construction for asymptotic shapes arising from finite initial seeds. Such a
shape converges to the polygonal invariant shape of the corresponding
deterministic model as the perturbation decreases. In many cases, exact
stability is observed. That is, for small perturbations, the shapes of the
deterministic and random processes agree exactly. We give a complete
characterization of such cases, and show that they are prevalent among
threshold growth CA with box neighborhood. We also design a nontrivial family
of CA in which the shape is exactly computable for all values of its
probability parameter.Comment: Published at http://dx.doi.org/10.1214/009117905000000512 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Slow Convergence in Bootstrap Percolation
In the bootstrap percolation model, sites in an L by L square are initially
infected independently with probability p. At subsequent steps, a healthy site
becomes infected if it has at least 2 infected neighbours. As
(L,p)->(infinity,0), the probability that the entire square is eventually
infected is known to undergo a phase transition in the parameter p log L,
occurring asymptotically at lambda = pi^2/18. We prove that the discrepancy
between the critical parameter and its limit lambda is at least Omega((log
L)^(-1/2)). In contrast, the critical window has width only Theta((log
L)^(-1)). For the so-called modified model, we prove rigorous explicit bounds
which imply for example that the relative discrepancy is at least 1% even when
L = 10^3000. Our results shed some light on the observed differences between
simulations and rigorous asymptotics.Comment: 22 pages, 3 figure
Percolation on fitness landscapes: effects of correlation, phenotype, and incompatibilities
We study how correlations in the random fitness assignment may affect the
structure of fitness landscapes. We consider three classes of fitness models.
The first is a continuous phenotype space in which individuals are
characterized by a large number of continuously varying traits such as size,
weight, color, or concentrations of gene products which directly affect
fitness. The second is a simple model that explicitly describes
genotype-to-phenotype and phenotype-to-fitness maps allowing for neutrality at
both phenotype and fitness levels and resulting in a fitness landscape with
tunable correlation length. The third is a class of models in which particular
combinations of alleles or values of phenotypic characters are "incompatible"
in the sense that the resulting genotypes or phenotypes have reduced (or zero)
fitness. This class of models can be viewed as a generalization of the
canonical Bateson-Dobzhansky-Muller model of speciation. We also demonstrate
that the discrete NK model shares some signature properties of models with high
correlations. Throughout the paper, our focus is on the percolation threshold,
on the number, size and structure of connected clusters, and on the number of
viable genotypes.Comment: 31 pages, 4 figures, 1 tabl
A sharper threshold for bootstrap percolation in two dimensions
Two-dimensional bootstrap percolation is a cellular automaton in which sites
become 'infected' by contact with two or more already infected nearest
neighbors. We consider these dynamics, which can be interpreted as a monotone
version of the Ising model, on an n x n square, with sites initially infected
independently with probability p. The critical probability p_c is the smallest
p for which the probability that the entire square is eventually infected
exceeds 1/2. Holroyd determined the sharp first-order approximation: p_c \sim
\pi^2/(18 log n) as n \to \infty. Here we sharpen this result, proving that the
second term in the expansion is -(log n)^{-3/2+ o(1)}, and moreover determining
it up to a poly(log log n)-factor. The exponent -3/2 corrects numerical
predictions from the physics literature.Comment: 21 page
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