Related papers: First passage percolation and a model for competin…
Random growth models are fundamental objects in modern probability theory, have given rise to new mathematics, and have numerous applications, including tumor growth and fluid flow in porous media. In this article, we introduce some of the…
We examine the two-dimensional extension of the model of Kessler and Sander of competition between two species identical except for dispersion rates. In this class of models, the spatial inhomogeneity of reproduction rates gives rise to an…
We consider standard first-passage percolation on $\Z^d$. Let $e_1$ be the first coordinate vector. Let $a(n)$ be the expected passage time from the origin to $ne_1$. In this short paper, we note that $a(n)$ is increasing under some strong…
We investigate the evolution of competing languages, a subject where much previous literature suggests that the outcome is always the domination of one language over all the others. Since coexistence of languages is observed in reality, we…
We consider the first passage percolation model in Z2 with a distribution F for 0 < F (0) < pc. In this paper, we solve the height problem.
Formation and competition of associations are studied in a six-species ecological model where each species has two predators and two prey. Each site of a square lattice is occupied by an individual belonging to one of the six species. The…
In first-passage percolation, one places nonnegative i.i.d. random variables (T (e)) on the edges of Z d. A geodesic is an optimal path for the passage times T (e). Consider a local property of the time environment. We call it a pattern. We…
Spatial self-organization emerges in distributed systems exhibiting local interactions when nonlinearities and the appropriate propagation of signals are at work. These kinds of phenomena can be modeled with different frameworks, typically…
Local coexistence of species in large ecosystems is traditionally explained within the broad framework of niche theory. However, its rationale hardly justifies rich biodiversity observed in nearly homogeneous environments. Here we consider…
We include alignment interactions in a well-studied first-order attractive-repulsive macroscopic model for aggregation. The distinctive feature of the extended model is that the equation that specifies the velocity in terms of the…
The site percolation on the triangular lattice stands out as one of the few exactly solved statistical systems. By initially configuring critical percolation clusters of this model and randomly reassigning the color of each percolation…
Time evolutions of number of cities, population of cities, world population, and size distribution of present languages are studied in terms of a new model, where population of each city increases by a random rate and decreases by a random…
We develop a model to compute the first-passage time of a random walker in a crowded environment. Hard-core particles with the same size and diffusion coefficient than the tracer diffuse, and the model allows to compute the first passage…
Environmental heterogeneity can drive genetic heterogeneity in expanding populations; mutant strains may emerge that trade overall growth rate for an improved ability to survive in patches that are hostile to the wild type. This…
Many real world networks have groups of similar nodes which are vulnerable to the same failure or adversary. Nodes can be colored in such a way that colors encode the shared vulnerabilities. Using multiple paths to avoid these…
Percolation plays an important role in fields and phenomena as diverse as the study of social networks, the dynamics of epidemics, the robustness of electricity grids, conduction in disordered media, and geometric properties in statistical…
Spatial pattern formation is a key feature of many natural systems in physics, chemistry and biology. The essential theoretical issue in understanding pattern formation is to explain how a spatially homogeneous initial state can undergo…
We study first-passage percolation where edges in the left and right half-planes are assigned values according to different distributions. We show that the asymptotic growth of the resulting inhomogeneous first-passage process obeys a shape…
We study a geometrically constrained coalescence model derived from spin systems. Given two probability distributions $\mathbb{P}_R$ and $\mathbb{P}_B$ on the positive reals with finite means, colour the real line alternately with red and…
We study the model theory of expansions of Hilbert spaces by generic predicates. We first prove the existence of model companions for generic expansions of Hilbert spaces in the form first of a distance function to a random substructure,…