The cone percolation on $\bbT_d$

TL;DR

This paper investigates a rumor spreading model on an infinite tree using percolation theory, providing bounds for the probability of widespread dissemination based on individuals' influence radii.

Contribution

It introduces bounds for the probability of a rumor reaching infinitely many individuals, connecting percolation theory with branching processes in a novel way.

Findings

Derived sharp bounds for the probability of infinite rumor spread.

Linked the existence of a giant component to the distribution of influence radii.

Provided conditions under which the rumor percolates infinitely.

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.