= Dimension comparison for the contact process
{title2=$\theta_d(\lambda)\geq\theta_1(d\lambda),\quad\lambda_c(d)\leq\lambda_c(1)/d$}
Map a site $x\in\mathbb Z^d$ to its height $h(x)=\sum_i x_i$. Maintain one infected representative in height $n$ for every infected site of an auxiliary one-dimensional <contact process>. The representative's death makes that auxiliary site recover. Its $d$ outgoing arrows to each neighbouring height, each of rate $\lambda$, make an auxiliary infection at total rate $d\lambda$, using the arrow's target as the new representative. Arrows to an already infected auxiliary height do nothing there. Distinct active representatives use distinct death and outgoing-arrow clocks, so their conditional rates are exactly those of the one-dimensional process. Every auxiliary infection has a live representative upstairs, proving the survival comparison.
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