= Height-weighted local extinction bound for the contact process
{title2=$\lambda_2\ge1/(2\sqrt d)$}
Orient a degree-$(d+1)$ <regular tree> towards one end, and let height $g$ decrease along the unique parent <edge> and increase along the $d$ child <edges>. For $w_\rho(A)=\sum_{x\in A}\rho^{g(x)}$, suppressing the occupied-target constraint only increases births, so its <infinitesimal generator> satisfies $Lw_\rho\le[-1+\lambda(\rho^{-1}+d\rho)]w_\rho$. For $d>1$, choose $\rho=d^{-1/2}$. If $\lambda<1/(2\sqrt d)$, the expected weight and the infection <probability> at each fixed site decay exponentially. The expected number of recoveries there equals its expected total occupation time, which is finite. Each infection episode ends almost surely at a recovery, so infinitely late infection is impossible. For $d=1$, take $\rho<1$ sufficiently close to one for each $\lambda<1/2$.
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