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We consider a dynamical process on a graph $G$, in which vertices are infected (randomly) at a rate which depends on the number of their neighbours that are already infected. This model includes bootstrap percolation and first-passage percolation as its extreme points. We give a precise description of the evolution of this process on the graph $mathbb{Z}^2$, significantly sharpening results of Dehghanpour and Schonmann. In particular, we determine the typical infection time up to a constant factor for almost all natural values of the parameters, and in a large range we obtain a stronger, sharp threshold.
We study survival among two competing types in two settings: a planar growth model related to two-neighbour bootstrap percolation, and a system of urns with graph-based interactions. In the planar growth model, uncoloured sites are given a colour at
We consider an evolving preferential attachment random graph model where at discrete times a new node is attached to an old node, selected with probability proportional to a superlinear function of its degree. For such schemes, it is known that the g
In this article, we try to give an answer to the simple question: ``textit{What is the critical growth rate of the dimension $p$ as a function of the sample size $n$ for which the Central Limit Theorem holds uniformly over the collection of $p$-dimen
Nacre is a layered, iridescent lining found inside many mollusk shells, with a unique brick-and-mortar periodic structure at the sub-micron scale, and remarkable resistance to fracture. Despite extensive studies, it remains unclear how nacre forms. H
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 e