Metastable Behavior of Bootstrap Percolation on Galton-Watson Trees

We consider Bernoulli bond percolation on a large scale-free tree in the supercritical regime, meaning informally that there exists a giant cluster with high probability. We obtain a weak limit theorem for the sizes of the next largest clusters, extending a recent result for large random recursive trees. The approach relies on the analysis of the asymptotic behavior of branching processes subject to rare neutral mutations, which may be of independent interest.

In this paper we analyze several anisotropic bootstrap percolation models in three dimensions. We present the order of magnitude for the metastability threshold for a fairly general class of models. In our proofs we use an adaptation of the technique of dimensional reduction. We find that the order of the metastability threshold is generally determined by the "easiest growth direction" in the model. In contrast to the anisotropic bootstrap percolation in two dimensions, in th...

Bootstrap percolation provides an emblematic instance of phase behavior characterised by an abrupt transition with diverging critical fluctuations. This unusual hybrid situation generally occurs in particle systems in which the occupation probability of a site depends on the state of its neighbours through a certain threshold parameter. In this paper we investigate the phase behavior of the bootstrap percolation on the regular random graph in the limit in which the threshold ...

We study the distribution of the percolation time $T$ of two-neighbour bootstrap percolation on $[n]^2$ with initial set $A\sim\mathrm{Bin}([n]^2,p)$. We determine $T$ with high probability up to a constant factor for all $p$ above the critical probability for percolation, and to within a $1+o(1)$ factor for a large range of $p$.

We study biased random walk on subcritical and supercritical Galton-Watson trees conditioned to survive in the transient, sub-ballistic regime. By considering offspring laws with infinite variance, we extend previously known results for the walk on the supercritical tree and observe new trapping phenomena for the walk on the subcritical tree which, in this case, always yield sub-ballisticity. This is contrary to the walk on the supercritical tree which always has some ballist...

We consider an interacting particle system on trees known as the frog model: initially, a single active particle begins at the root and i.i.d.~$\mathrm{Poiss}(\lambda)$ many inactive particles are placed at each non-root vertex. Active particles perform discrete time simple random walk and activate the inactive particles they encounter. We show that for Galton-Watson trees with offspring distributions $Z$ satisfying $\mathbf{P}(Z \geq 2) = 1$ and $\mathbf{E}[Z^{4 + \epsilon}]...

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 dimensions, termed subcritical. We introduce the new notion of `critical densities' serving the role of `difficulties' for critical models, but adapted to subcritical ones. We characterise the critical probability in terms of these quantities a...

Bootstrap percolation has been used effectively to model phenomena as diverse as emergence of magnetism in materials, spread of infection, diffusion of software viruses in computer networks, adoption of new technologies, and emergence of collective action and cultural fads in human societies. It is defined on an (arbitrary) network of interacting agents whose state is determined by the state of their neighbors according to a threshold rule. In a typical setting, bootstrap per...

Bootstrap percolation transition may be first order or second order, or it may have a mixed character where a first order drop in the order parameter is preceded by critical fluctuations. Recent studies have indicated that the mixed transition is characterized by power law avalanches, while the continuous transition is characterized by truncated avalanches in a related sequential bootstrap process. We explain this behavior on the basis of a through analytical and numerical st...

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 probability $p$ between any pair of vertices. Bootstrap percolation on a random graph is a process of spread of "activation" on a given realisation of the graph with a given number of initially active nodes. At each step those vertices which have ...