Related papers: Optimal and Near-Optimal Constructions for Bootstr…

This paper is dedicated to the study of the interaction between dynamical systems and percolation models, with views towards the study of viral infections whose virus mutate with time. Recall that r-bootstrap percolation describes a…

The process of $H$-bootstrap percolation for a graph $H$ is a cellular automaton, where, given a subset of the edges of $K_n$ as initial set, an edge is added at time $t$ if it is the only missing edge in a copy of $H$ in the graph obtained…

A 1-factorisation of a regular graph $G$ is a partition of its edge set $E(G)$ into perfect matchings of $G$. Behague asked for the minimal $r=r(d)$ such that some $1$-factorisation of the $d$-dimensional hypercube $Q_d$ has the property…

Bootstrap percolation on the random graph $G_{n,p}$ is a process of spread of "activation" on a given realization of the graph with a given number of initially active nodes. At each step those vertices which have not been active but have at…

We prove that nearest-neighbor percolation in dimensions $d\geq 11$ displays mean-field behavior by proving that the infrared bound holds, in turn implying the finiteness of the percolation triangle diagram. The finiteness of the triangle…

The Hamming graph $H(d,n)$ is the Cartesian product of $d$ complete graphs on $n$ vertices. Let $m=d(n-1)$ be the degree and $V = n^d$ be the number of vertices of $H(d,n)$. Let $p_c^{(d)}$ be the critical point for bond percolation on…

A graph $G$ is $m$-minor-universal if every graph with at most $m$ edges (and no isolated vertices) is a minor of $G$. We prove that the $d$-dimensional hypercube, $Q_d$, is $\Omega\left(\frac{2^d}{d}\right)$-minor-universal, and that there…

In the first paper of this series [S. Torquato, J. Chem. Phys. {\bf 136}, 054106 (2012)], analytical results concerning the continuum percolation of overlapping hyperparticles in $d$-dimensional Euclidean space $\mathbb{R}^d$ were obtained,…

In modified two-neighbour bootstrap percolation in two dimensions each site of $\mathbb Z^2$ is initially independently infected with probability $p$ and on each discrete time step one additionally infects sites with at least two…

Bootstrap percolation is a wide class of monotone cellular automata with random initial state. In this work we develop tools for studying in full generality one of the three `universality' classes of bootstrap percolation models in two…

We study percolation problems of overlapping objects where the underlying geometry is such that in D-dimensions, a subset of the directions has a lattice structure, while the remaining directions have a continuum structure. The resulting…

In the modified bootstrap percolation model, sites in the cube {1,...,L}^d are initially declared active independently with probability p. At subsequent steps, an inactive site becomes active if it has at least one active nearest neighbour…

An irreversible $k$-threshold process (also a $k$-neighbor bootstrap percolation) is a dynamic process on a graph where vertices change color from white to black if they have at least $k$ black neighbors. An irreversible $k$-conversion set…

Let $\mathbb{G}=\left(\mathbb{V},\mathbb{E}\right)$ be the graph obtained by taking the cartesian product of an infinite and connected graph $G=(V,E)$ and the set of integers $\mathbb{Z}$. We choose a collection $\mathcal{C}$ of finite…

We study Voronoi percolation on a large class of $d$-dimensional Riemannian manifolds, which includes the hyperbolic spaces $\mathbb{H}^d$, $d\geq 2$. We prove that as the intensity $\lambda$ of the underlying Poisson point process tends to…

By definition, a rigid graph in $\mathbb{R}^d$ (or on a sphere) has a finite number of embeddings up to rigid motions for a given set of edge length constraints. These embeddings are related to the real solutions of an algebraic system.…

Let H = (H,V) be a hypergraph with edge set H and vertex set V. Then hypergraph H is invertible iff there exists a permutation pi of V such that for all E belongs to H(edges) intersection of(pi(E) and E)=0. H is invertibility critical if H…

Equip each point $x$ of a homogeneous Poisson process $\mathcal{P}$ on $\mathbb{R}$ with $D_x$ edge stubs, where the $D_x$ are i.i.d. positive integer-valued random variables with distribution given by $\mu$. Following the stable…

In \emph{$k$-bootstrap percolation}, we fix $p\in (0,1)$, an integer $k$, and a plane graph $G$. Initially, we infect each face of $G$ independently with probability $p$. Infected faces remain infected forever, and if a healthy (uninfected)…

We define a graph process $\mathcal{G}(p,q)$ based on a discrete branching process with deletions and mergers, which is inspired by the 4-cycle structure of both the hypercube $Q_d$ and the lattice $\mathbb{Z}^d$ for large $d$. Individuals…