Percolation of the contact process on the regular tree

Abstract: The contact process on the regular tree $\mathbb{T}<em>d$ when $d\geq 3$ has the two phase transitions of global and of local survival, found by Pemantle and Liggett at values $λ_1$ and $λ_2$. 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, $λ_p$ is the critical value beyond which the infected vertices can percolate through $\mathbb{T}_d$, and $λ</em>{p<sup>\complement}$ is the parameter before which the healthy vertices can percolate. We find these are both distinct phase transitions [0<λ1<λ_p<λ_2<λ{p^\complement}<+\infty] on $\mathbb{T}_d$ when $d\geq 7$. The most interesting of these comparisons is $λ_1&lt;λ_p$, which we find for all $d\geq 3$. This comparison $λ_1&lt;λ_p$ is a long-standing open question on $\mathbb{Z}<sup>d$ with $d\geq 2$ and was not yet found on any other graphs except where $λ_p$ is infinite.