Percolation of the contact process on the regular tree

The contact process on the regular tree ${\mathbb{T}}_d$ when $d\ge 3$ has the two phase transitions of global and of local survival, found by Pemantle and Liggett at values ${\lambda }_1$ and ${\lambda }_2$. We start the system with every vertex infected and let it relax to what is known as the upper invariant infection. In this stationary state, ${\lambda }_p$ is the critical value beyond which the infected vertices can percolate through ${\mathbb{T}}_d$, and ${\lambda }_{p^{\mathrm{\complement }}}$ is the parameter before which the healthy vertices can percolate. We find these are both distinct phase transitions $$0<{\lambda }_1<{\lambda }_p<{\lambda }_2<{\lambda }_{p^{\mathrm{\complement }}}<+\mathrm{\infty }$$ on ${\mathbb{T}}_d$ when $d\ge 7$. The most interesting of these comparisons is ${\lambda }_1<{\lambda }_p$, which we find for all $d\ge 3$. This comparison ${\lambda }_1<{\lambda }_p$ is a long-standing open question on ${\mathbb{Z}}^d$ with $d\ge 2$ and was not yet found on any other graphs except where ${\lambda }_p$ is infinite.