Critical Percolation and the Incipient Infinite Cluster on Galton-Watson Trees

We consider critical percolation on a supercritical Galton-Watson tree. We show that, when the offspring distribution is in the domain of attraction of an $\alpha$-stable law for some $\alpha \in (1,2)$, or has finite variance, several annealed properties also hold in a quenched setting. In particular, the following properties hold for the critical root cluster on almost every realisation of the tree: (1) the rescaled survival probabilities converge; (2) the Yaglom limit or i...

We study survival properties of inhomogeneous Galton-Watson processes. We determine the so-called branching number (which is the reciprocal of the critical value for percolation) for these random trees (conditioned on being infinite), which turns out to be an a.s.\ constant. We also shed some light on the way the survival probability varies between the generations. When we perform independent percolation on the family tree of an inhomogeneous Galton-Watson process, the result...

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.

We explore the survival function for percolation on Galton-Watson trees. Letting $g(T,p)$ represent the probability a tree $T$ survives Bernoulli percolation with parameter $p$, we establish several results about the behavior of the random function $g(\mathbf{T} , \cdot)$, where $\mathbf{T}$ is drawn from the Galton-Watson distribution. These include almost sure smoothness in the supercritical region; an expression for the $k\text{th}$-order Taylor expansion of $g(\mathbf{T} ...

Bootstrap percolation is a cellular automaton modelling the spread of an `infection' on a graph. In this note, we prove a family of lower bounds on the critical probability for $r$-neighbour bootstrap percolation on Galton--Watson trees in terms of moments of the offspring distributions. With this result we confirm a conjecture of Bollob\'as, Gunderson, Holmgren, Janson and Przykucki. We also show that these bounds are best possible up to positive constants not depending on t...

We study oriented percolation on random causal triangulations, those are random planar graphs obtained roughly speaking by adding horizontal connections between vertices of an infinite tree. When the underlying tree is a geometric Galton--Watson tree with mean $m>1$, we prove that the oriented percolation undergoes a phase transition at $p_c(m)$, where $p_c(m) = \frac{\eta}{1+\eta}$ with $\eta = \frac{1}{m+1} \sum_{n \geq 0} \frac{m-1}{m^{n+1}-1}$. We establish that strictly ...

Bootstrap percolation is a type of cellular automaton which has been used to model various physical phenomena, such as ferromagnetism. For each natural number $r$, the $r$-neighbour bootstrap process is an update rule for vertices of a graph in one of two states: `infected' or `healthy'. In consecutive rounds, each healthy vertex with at least $r$ infected neighbours becomes itself infected. Percolation is said to occur if every vertex is eventually infected. Usually, the s...

We consider invasion percolation on Galton-Watson trees. On almost every Galton-Watson tree, the invasion cluster almost surely contains only one infinite path. This means that for almost every Galton-Watson tree, invasion percolation induces a probability measure on infinite paths from the root. We show that under certain conditions of the progeny distribution, this measure is absolutely continuous with respect to the limit uniform measure. This confirms that invasion percol...

We analyse the cluster discovered by invasion percolation on a branching process with a power-law offspring distribution. Invasion percolation is a paradigm model of self-organised criticality, where criticality is approached without tuning any parameter. By performing invasion percolation for $n$ steps, and letting $n\to\infty$, we find an infinite subtree, called the invasion percolation cluster (IPC). A notable feature of the IPC is its geometry that consists of a unique p...

Let $\mathcal{T}$ be a supercritical Galton-Watson tree with a bounded offspring distribution that has mean $\mu >1$, conditioned to survive. Let $\varphi_{\mathcal{T}}$ be a random embedding of $\mathcal{T}$ into $\mathbb{Z}^d$ according to a simple random walk step distribution. Let $\mathcal{T}_p$ be percolation on $\mathcal{T}$ with parameter $p$, and let $p_c = \mu^{-1}$ be the critical percolation parameter. We consider a random walk $(X_n)_{n \ge 1}$ on $\mathcal{T}_p$...