Graphical representation of the contact process

Place recovery marks at each vertex according to independent rate-$\mu$ Poisson processes and infection arrows on each oriented nearest-neighbour edge according to independent rate-$\lambda$ Poisson processes. A vertex is infected at time $t$ exactly when a time-directed path from an initially infected vertex reaches it while following arrows and avoiding recovery marks.

For the one-dimensional contact process with recovery rate one and per-neighbour infection rate $\lambda$, use parity sites $(x,n)$ with $x+n$ even. Call a site good if $x$ has no recovery in $[(n-1)\delta ,(n+1)\delta ]$ and sends an arrow to each neighbour during $[n\delta ,(n+1)\delta ]$. The goodness indicators are independent: recovery intervals for successive parity sites at the same spatial point only touch at endpoints, and all outgoing-arrow intervals and clocks are disjoint. Their common probability is the displayed $q$. A path of good sites carries infection from each time layer to the next, because both its source and target avoid recovery during the transmitting interval. Taking $\delta ={\lambda }^{-1/2}$ makes $q\to 1$. The incoming-edge coupling of site and bond percolation and $p_c^{\mathrm{bond}}<1$ imply $p_c^{\mathrm{site}}<1$, so this proves a finite critical infection rate.

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.

Reachability in the graphical representation gives

Time reversal of the Poisson graphical representation, together with reversal of every infection arrow, gives