Related papers: Noise sensitivity in bootstrap percolation

We prove that the Poisson Boolean model, also known as the Gilbert disc model, is noise sensitive at criticality. This is the first such result for a Continuum Percolation model, and the first for which the critical probability p_c \ne 1/2.…

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…

Consider the problem of determining the maximal induced subgraph in a random $d$-regular graph such that its components remain bounded as the size of the graph becomes arbitrarily large. We show, for asymptotically large $d$, that any such…

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 consider the $r$-neighbor bootstrap percolation process on the graph with vertex set $V=\{0,1\}^n$ and edges connecting the pairs at Hamming distance $1,2,\dots,k$, where $k\ge 2$. We find asymptotics of the critical probability of…

Two-dimensional bootstrap percolation is usually characterized by bulk observables, but whether increasing the activation threshold qualitatively reorganizes the geometry of the absorbing state has remained unclear. Here we show that the…

Consider a $p$-random subset $A$ of initially infected vertices in the discrete cube $[L]^d$, and assume that the neighbourhood of each vertex consists of the $a_i$ nearest neighbours in the $\pm e_i$-directions for each $i \in \{1,2,\dots,…

Let $G_{n,p}^1$ be a superposition of the random graph $G_{n,p}$ and a one-dimensional lattice: the $n$ vertices are set to be on a ring with fixed edges between the consecutive vertices, and with random independent edges given with…

Bootstrap percolation is a prominent framework for studying the spreading of activity on a graph. We begin with an initial set of active vertices. The process then proceeds in rounds, and further vertices become active as soon as they have…

In this paper we fill in a fundamental gap in the extremal bootstrap percolation literature, by providing the first proof of the fact that for all $d \geq 1$, the size of the smallest percolating sets in $d$-neighbour bootstrap percolation…

Bootstrap percolation on a graph with infection threshold $r\in \mathbb{N}$ is an infection process, which starts from a set of initially infected vertices and in each step every vertex with at least $r$ infected neighbours becomes…

Consider a graph $G$ and an initial random configuration, where each node is black with probability $p$ and white otherwise, independently. In discrete-time rounds, each node becomes black if it has at least $r$ black neighbors and white…

Bootstrap percolation on an arbitrary graph has a random initial configuration, where each vertex is occupied with probability p, independently of each other, and a deterministic spreading rule with a fixed parameter k: if a vacant site has…

We consider random walk on a mildly random environment on finite transitive d- regular graphs of increasing girth. After scaling and centering, the analytic spectrum of the transition matrix converges in distribution to a Gaussian noise. An…

It is shown that a large class of events in a product probability space are highly sensitive to noise, in the sense that with high probability, the configuration with an arbitrary small percent of random errors gives almost no prediction…

We study two-dimensional critical bootstrap percolation models. We establish that a class of these models including all isotropic threshold rules with a convex symmetric neighbourhood, undergoes a sharp metastability transition. This…

We examine bootstrap percolation in d-dimensional, directed metric graphs in the context of recent measurements of firing dynamics in 2D neuronal cultures. There are two regimes, depending on the graph size N. Large metric graphs are…

Bootstrap percolation models have been extensively studied during the two past decades. In this article, we study the following "anisotropic" bootstrap percolation model: the neighborhood of a point (m,n) is the set…

Percolation theory has become a useful tool for the analysis of large-scale wireless networks. We investigate the fundamental problem of characterizing the critical density $\lambda_c^{(d)}$ for $d$-dimensional Poisson random geometric…

In majority bootstrap percolation on a graph G, an infection spreads according to the following deterministic rule: if at least half of the neighbours of a vertex v are already infected, then v is also infected, and infected vertices remain…