= Branching bound for contact-process survival
{title2=$\lambda_c(d)\geq(2d)^{-1}$}
With recovery rate one, the number of infected sites in a nearest-neighbour <contact process> on $\mathbb Z^d$ has death rate equal to its size and birth rate at most $2d\lambda$ times its size. It is dominated by a linear birth-death <branching process> with these per-particle rates. In particular, from one infection its <expected value>[expected size] is at most $e^{(2d\lambda-1)t}$, so its <Survival probability> is zero when $2d\lambda<1$. This proves the displayed lower bound on the critical rate.
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