We also show that this bound is sharp up to a factor of O(b) by giving an explicit family of Galton–Watson trees with critical probability bounded from above by some constant Cr > 0. Percolation On Galton-Watson Trees Abstract We consider both Bernoulli and invasion percolation on Galton-Watson trees. Watson generation sizes as a sum of independent increments which is derived from the decomposition of the conditioned Galton-Watson family tree along the. ![]() Probability pc(T, r) 0 such that if T is a Galton–Watson tree with branching number br(T) = b ≥ r then pc > f(b,cr, r). Such a conditioned Galton-Watson tree will be abbreviated as CGW (n)-tree. In this paper, we look at infinite trees and, answering a problem posed by Balogh, Peres and Pete, we show that for any b ≥ r and for any > 0 there exists a tree T with branching number br(T) = b and critical In that case, given a graph G and infection threshold r, a quantity of interest is the critical probability, pc(G, r), at which percolation becomes likely to occur. Informally, the associated cut-tree represents the genealogy of the nested connected components created by this process. Usually, the starting set of infected vertices is chosen at random, with all vertices initially infected independently with probability p. If q 0 we recover a usual Galton-Watson tree and the survival threshold for the process has p > 0. random potential whose marginal distribution is double-exponential. We introduce a generalization of Galton-Watson trees where, individuals have independently a number of Poi(1 p) offspring and, at each generation, pairs of cousins merge independently with probability q. Percolation is said to occur if every vertex is eventually infected. Progress in probability, Birkhuser, Basel, 2021) a detailed analysis was given of the large-time asymptotics of the total mass of the solution to the parabolic Anderson model on a supercritical GaltonWatson random tree with an i.i.d. ![]() In consecutive rounds, each healthy vertex with at least r infected neighbours becomes itself infected. 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’. Bootstrap percolation is a type of cellular automaton which has been used to model various physical phenomena, such as ferromagnetism. Key words: Random Galton-Watson tree, probabilistic analysis of algorithms, branching process.
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