Critical branching-rate conjecture for coexistence in the symbiotic branching model

Let $d\geq 3$ and let $\varrho>0$. In the two-type symbiotic branching model $\textrm{SBM}_\gamma(\varrho)$ on $\mathbb{Z}^d$, coexistence means that there are summable initial conditions $u_0,v_0$ such that

$$\langle u_\infty,\1\rangle\langle v_\infty,\1\rangle>0$$

with positive probability. Critical branching-rate conjecture. There is a critical constant $\gamma(\varrho,d)\in(0,\infty)$ such that coexistence occurs if $\gamma<\gamma(\varrho,d)$ and coexistence is impossible if $\gamma>\gamma(\varrho,d)$. The recurrence/transience question for positive correlation in dimensions $d\geq 3$ remains open; the preceding results establish the corresponding dichotomy for zero correlation and for negative correlation.