Related papers: First passage percolation and a model for competin…
We study first-passage percolation in two dimensions, using measures mu on passage times with b:=inf supp(mu) >0 and mu({b})=p \geq p_c, the threshold for oriented percolation. We first show that for each such mu, the boundary of the limit…
We generalize first-species counterpoint theory to arbitrary rings and obtain some new counting and maximization results that enrich the theory of admitted successors, pointing to a structural approach, beyond computations. The…
We introduce and solve a model which considers two coupled networks growing simultaneously. The dynamics of the networks is governed by the new arrival of network elements (nodes) making preferential attachments to pre-existing nodes in…
We uncover a duality between relaxation and first passage processes in ergodic reversible Markovian dynamics in both discrete and continuous state-space. The duality exists in the form of a spectral interlacing -- the respective time scales…
We propose a general theory for surface patterning in many different biological systems, including mite and insect cuticles, pollen grains, fungal spores, and insect eggs. The patterns of interest are often intricate and diverse, yet an…
In this work, we consider a diffusive two-species d-dimensional model and study it in great details. Two types of particles, with hard-core, diffuse symmetrically and cross each other. For arbitrary dimensions, we obtain the exact density,…
We consider first passage percolation on sparse random graphs with prescribed degree distributions and general independent and identically distributed edge weights assumed to have a density. Assuming that the degree distribution satisfies a…
We introduce the notion of crossings and nestings of a permutation. We compute the generating function of permutations with a fixed number of weak exceedances, crossings and nestings. We link alignments and permutation patterns to these…
We consider a Brownian particle diffusing in a one dimensional interval with absorbing end points. We study the ramifications when such motion is interrupted and restarted from the same initial configuration. We provide a comprehensive…
We consider the genealogy of a sample of individuals taken from a spatially structured population when the variance of the offspring distribution is relatively large. The space is structured into discrete sites of a graph G. If the…
This paper concerns the intersection of natural language and the physical space around us in which we live, that we observe and/or imagine things within. Many important features of language have spatial connotations, for example, many…
A model of directed percolation processes with colors and flavors that is equivalent to a population model with many species near their extinction thresholds is presented. We use renormalized field theory and demonstrate that all…
We study a particular model of a random medium, called the orthant model, in general dimensions $d\ge 2$. Each site $x\in \Z^d$ independently has arrows pointing to its positive neighbours $x+e_i$, $i=1,\dots, d$ with probability $p$ and…
Consider a graph where the sites are distributed in space according to a Poisson point process on $\mathbb R^n$. We study a population evolving on this network, with individuals jumping between sites with a rate which decreases…
Resource competition is a fundamental interaction in natural communities.However little is known about competition in spatial environments where organisms are able to regulate resource distributions. Here, we analyze the competition of two…
A rapidly increasing number of systems is identified in which the stochastic motion of tracer particles follows the Brownian law $\langle\mathbf{r}^2(t) \rangle\simeq Dt$ yet the distribution of particle displacements is strongly…
We generalize the standard site percolation model on the $d$-dimensional lattice to a model on random tessellations of $\mathbb R^d$. We prove the uniqueness of the infinite cluster by adapting the Burton-Keane argument…
It has been suggested that major transitions in evolution require the emergence of novelties, often associated to the cooperative behaviour of previously existing objects or agents. A key innovation involves the first cooperative…
We consider the first passage percolation model on the square lattice with an edge weight distribution F. In this paper, we consider the number of optimal paths for two points separated by a long distance. We show that there is a phase…
Understanding how species persist under interacting stressors is a central challenge in ecology. We develop a spatially explicit reaction-diffusion framework to investigate competing species in landscapes shaped by climate variability,…