Survival-regime conjecture for competing first passage percolation on lattices

Consider the competing first passage percolation model with conversion on $\mathbb{Z}^d$, where $d\geq2$, and let $\lambda$ be the infection rate relative to awareness and $\rho$ the conversion rate.

Survival-regime conjecture.

• For all $\lambda<1$, if $\rho$ is sufficiently small, type 1 survives with positive probability.
• There exists $\rho_c=\rho_c(d)\in(0,\infty)$ such that:

1. if $\rho<\rho_c$ and $\lambda$ is sufficiently small, type 1 survives with positive probability;
2. if $\rho>\rho_c$ and $\lambda>0$, then type 1 dies out almost surely.

A complete picture of the survival regimes on $\mathbb{Z}^d$ is open. Together with the preceding lattice theorem, this conjecture describes the expected phase diagram for survival and extinction of type 1 as the infection, awareness, and conversion rates vary.