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

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...

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...

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 $\mathcal{T}_p$...

In dynamical percolation, the status of every bond is refreshed according to an independent Poisson clock. For graphs which do not percolate at criticality, the dynamical sensitivity of this property was analyzed extensively in the last decade. Here we focus on graphs which percolate at criticality, and investigate the dynamical sensitivity of the infinite cluster. We first give two examples of bounded degree graphs, one which percolates for all times at criticality and one w...

We study Bernoulli bond percolation on a random recursive tree of size $n$ with percolation parameter $p(n)$ converging to $1$ as $n$ tends to infinity. The sizes of the percolation clusters are naturally stored in a tree. We prove convergence in distribution of this tree to the genealogical tree of a continuous-state branching process in discrete time. As a corollary we obtain the asymptotic sizes of the largest and next largest percolation clusters, extending thereby a rece...

We analyze the metastable states near criticality of the bootstrap percolation on Galton-Watson trees. We find that, depending on the exact choice of the offspring distribution, it is possible to have several distinct metastable states, with varying scaling of their duration while approaching criticality.

When considering statistical mechanics models on trees, such that the Ising model, percolation, or more generally the random cluster model, some concave tree recursions naturally emerge. Some of these recursions can be compared with non-linear conductances, or $p$-conductances, between the root and the leaves of the tree. In this article, we estimate the $p$-conductances of $T_n$, a supercritical Galton-Watson tree of depth $n$, for any $p>1$ (for a quenched realization of $T...

We consider the simple random walk on the infinite cluster of the Bernoulli bond percolation of trees, and investigate the relation between the speed of the simple random walk and the retaining probability p by studying three classes of trees. A sufficient condition is established for Galton-Watson trees.

Consider independent bond percolation with retention probability p on a spherically symmetric tree Gamma. Write theta_Gamma(p) for the probability that the root is in an infinite open cluster, and define the critical value p_c=inf{p:theta_Gamma(p)>0}. If theta_Gamma(p_c)=0, then the root may still percolate in the corresponding dynamical percolation process at the critical value p_c, as demonstrated recently by Haggstrom, Peres and Steif. Here we relate this phenomenon to the...

In this article, we prove a joint large deviation principle in $n$ for the \emph{empirical pair measure} and \emph{ empirical offspring measure} of critical multitype Galton-Watson trees conditioned to have exactly $n$ vertices in the weak topology. From this result we extend the large deviation principle for the empirical pair measures of Markov chains on simply generated trees to cover offspring laws which are not treated by \cite[Theorem~2.1]{DMS03}. For the case where t...