The asymptotic critical branching-rate conjecture for the general contact process

Let $d\ge 3$ and let $P^{b}(x,y)$ satisfy the model's stated assumption. For the process, write

$$\lambda_c^N=\inf\{\lambda\ge0:\rho_\lambda^N>0\}$$

for the critical branching rate, and let

$$\vartheta=(2d)^{-1}\sum_{x,y\in {\mathbb{Z}^{d}}}P^{b}(0,x)P^{b}(0,y)G(x,y).$$

Critical branching-rate conjecture. In the general model,

$$\limsup_{N\to\infty}\frac{\lambda_c^N-1}{\frac{\vartheta}{N}}=1,$$

and hence

$$\lim_{N\to\infty}\frac{\lambda_c^N-1}{\frac{\vartheta}{N}}=1.$$

The theorem preceding the conjecture proves the matching lower bound in the liminf. The conjecture asserts that this lower bound is asymptotically sharp, determining the critical branching rate to first order as $1+\vartheta/N$; the corresponding upper bound is not established here.