Properties of the limit shape for some last passage growth models in random environments

We study directed last passage percolation on the first quadrant of the planar square lattice whose weights have general distributions, or equivalently, ./G/1 queues in series. The service time distributions of the servers vary randomly which constitutes a random environment for the model. Equivalently, each row of the last passage model has its own randomly chosen weight distribution. We investigate the limiting time constant close to the boundary of the quadrant. Close to the y-axis, where the number of random distributions averaged over stays large, the limiting time constant takes the same universal form as in the homogeneous model. But close to the x-axis we see the effect of the tail of the distribution of the random means attached to the rows.Comment: 24 pages, this paper has been accepted for publication in Stochastic Processes and their Application

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