The cone percolation on $\mathbb{T}_{d}$

Abstract

We study a rumour model from a percolation theory and branching process point of view. The existence of a giant component is related to the event where the rumour spreads out trough an infinite number of individuals. We present sharp lower and upper bounds for the probability of that event, according to the distribution of the random variables that defines the radius of influence of each individual.

Frog models on trees through renewal theory

“…These bounds improve the ones of [11] who obtained Table 1 for a comparison between (5) and Proposition 3.1.…”

“…To find a lower bound for q c , just observe that P q (A n ) = P q (∪ v:d(0,v)=n A v ) ≤ d n p q,n where p q,n denotes the common value (by symmetry) of the P q (A v )'s for any v at distance n of the root. Thus, (11) d n p q,n → 0 ⇒ q < q c .…”

“…Our interest is to compare this information propagation with the frog model on the directed tree, making c = 1 and q = p in (1). A quick look to the first step of the proof of Theorem 2.1 shows that (11) and (12) are also valid using the dynamic of the cone percolation in place of our frog model. The remaining of the proof only relies on the dynamic along one single branch, which is the same in both models with our choice of the parameters.…”

“…Cone percolation with geometric radius. The first application we discuss is an improvement of both, upper and lower bounds, for the critical probability of a long range percolation model on − → T d called cone percolation model, see [11]. There is a random variable X v associated to each vertex v ∈ V := V(T d ), representing a radius of propagation of a piece of information.…”

“…At time one all the vertices at distance of at most X 0 of the root of the tree are informed. At each step, each newly informed vertex v will inform all non-informed vertices v ′ > v such that [11] proved that there exists a critical value p cp c (the superscript "cp" stands for cone percolation) above which infinitely many vertices are informed (i.e., the model percolates) with positive probability.…”