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

Consider the Contact process started from a finite infected set $A$ on a degree-$(d+1)$ regular tree, with $d\ge 2$, recovery rate one and per-oriented-edge infection rate $\lambda >0$. Choose $0<\rho <1$ satisfying $\rho \lambda (d-1)\ge 1$. The bounded functional $f(A)={\rho }^{|A|}$ has generatorRecoveries decrease the infected count by one, whereas infections across the edge boundary increase it by one; summing these changes gives the formula.

A finite nonempty set induces a forest. If it has $c(A)$ connected components, edge counting gives $|{\partial }_{E}A|=(d-1)|A|+2c(A)$. Therefore the stated parameter condition makes $Lf\le 0$. At the absorbing empty state its generator is zero. Localizing the finite-particle process and using boundedness yields that ${\rho }^{|A_t|}$ is a supermartingale.

On extinction by time $t$, its value is one, so ${\mathbb{P}}_A(T_{\varnothing }\le t)\le {\mathbb{E}}_A{\rho }^{|A_t|}\le {\rho }^{|A|}$. Letting $t\to \infty$ proves the extinction bound ${\mathbb{P}}_A(T_{\varnothing }<\infty )\le {\rho }^{|A|}$. A strict choice $\rho <1$ exists when $\lambda (d-1)>1$, giving a sufficient survival condition from every nonempty finite seed. The range of $\rho$ is essential to both boundedness and the sign argument.