Exponential cardinality supermartingale for the contact process on a tree (${\nu }_{\rho }(A)={\rho }^{|A|}$)

For the contact process with per-edge infection rate $\lambda$ and recovery rate one on a degree-$(d+1)$ regular tree, the infinitesimal generator gives $L{\nu }_{\rho }(A)=(1-\rho ){\nu }_{\rho }(A)(|A|/\rho -\lambda |{\partial }_{E}A|)$. The edge boundary of a finite forest in a regular tree implies $L{\nu }_{\rho }\le 0$ whenever $\rho \lambda (d-1)\ge 1$, including the absorbing empty state where the generator is zero. Thus the displayed functional is a bounded supermartingale, and the extinction probability from $A$ is at most ${\rho }^{|A|}$.