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For the graphical representation of the contact process, put independent death marks on each vertical time-line $\{x\}\times [0,\infty )$ at rate $\delta$, and independent infection arrows $x\to y$ at rate $\lambda$ for every ordered nearest-neighbour pair. A time-directed path moves vertically upward, cannot pass a death mark, and may follow an infection arrow. From an initial set $A$, the infected set ${\xi }_t^A$ consists of sites reachable at time $t$ from $A\times \{0\}$. Thus infected sites recover at rate $\delta$, and a healthy site with $k$ infected neighbours becomes infected at rate $k\lambda$. Rates here are per ordered edge, fixing the normalization of $\lambda$.

With $\delta =1$, writeCouple two infection rates by adding independent arrows of rate ${\lambda }_2-{\lambda }_1$ to the rate-${\lambda }_1$ construction, retaining the same deaths. Every path available at the lower rate remains available at the higher rate. Hence ${\theta }_d(\lambda )$ is nondecreasing.

For the lower bound, dominate the infection population by a linear birth-death branching process $Z_t$, started from one particle, with per-particle birth rate $2d\lambda$ and death rate $1$. Assign one branching particle to every infection; allow an offspring at every attempted transmission, including attempts into an already infected site, and let the extra particles continue independently. Matched deaths remove their infection. This constructs $|{\xi }_t|\le Z_t$. The linear birth rate is nonexplosive, so in particular the contact process from a finite initial set is well-defined with finite population at finite times. Its branching mean solvesConsequentlyIf $2d\lambda <1$, let $t\to \infty$ to obtain ${\theta }_d(\lambda )=0$. This proves the branching bound for contact-process survival

For the upper bound use the dimension comparison for the contact process. Define the height $h(x)=x_1+\cdots +x_d$. Construct an auxiliary infected set ${\eta }_t\subseteq \mathbb{Z}$ together with an infected representative $r_n(t)\in {\xi }_t$ of height $n$ for every $n\in {\eta }_t$. Initially ${\eta }_0=\{0\}$ and $r_0=0$. Use the original graphical marks as follows.

This rule maintains a live representative for every auxiliary infection. Conditional on the full past, each active representative has death rate $1$ and outgoing-arrow rate $d\lambda$ in each of the two height directions. Representatives at distinct heights are different sites, using distinct death clocks and distinct outgoing ordered-edge clocks. Independence and memorylessness of the underlying Poisson processes give exactly the one-dimensional contact-process generatorThus $\eta$ has the law of the one-dimensional contact process with infection rate $d\lambda$, even though the representatives themselves depend on the higher-dimensional history. Since ${\eta }_t\ne \varnothing$ implies ${\xi }_t\ne \varnothing$ at every time,Every $\lambda$ for which $d\lambda >{\lambda }_c(1)$ is therefore supercritical whenever the one-dimensional survival set is nonempty; if that critical value were infinite the inequality would be trivial. Taking infima gives