Currently I am trying to understand the paper "Random Plane Networks" by E.N. Gilbert. In section 2 of this paper, he is deriving a lower bound for the expected number of points in the connection radius of a arbitrary point, $E$, for which infinite sized components start to emerge in infinite RGG's. He does this by defining a way of constructing a "more densely populated" component and calculating the probability that that component is infinite (which would yield an upper bound for the RGG case). In the paper he claims that his construction is related to a branching process found in the study of survival of families, and claims that if $E\leq 1$, then the probability that there is an infinite component is zero. I am having trouble understanding why this is this case. Any advice or references are appreciated, thanks in advance!



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