Percolation of the contact process on the regular tree

RESUMO

The contact process on the regular tree $\mathbb{T}_d$ when $d\geq 3$ 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 $\lambda_{p^\complement}$ is the parameter before which the healthy vertices can percolate. We find these are both distinct phase transitions \[0<\lambda_1<\lambda_p<\lambda_2<\lambda_{p^\complement}<+\infty\] on $\mathbb{T}_d$ when $d\geq 7$. The most interesting of these comparisons is $\lambda_1<\lambda_p$, which we find for all $d\geq 3$. This comparison $\lambda_1<\lambda_p$ is a long-standing open question on $\mathbb{Z}^d$ with $d\geq 2$ and was not yet found on any other graphs except where $\lambda_p$ is infinite.