A lower bound for the order parameter in the one-dimensional contact process (Q1272165)

A one-dimensional contact process with the parameter \(\lambda >0\) [as described e.g. in \textit{T. Liggett's} book ``Interacting particle systems'' (1985; Zbl 0559.60078)] is considered. It is known that there exists a critical value \(\lambda _{c}\in \left ]0,+\infty \right [\) such that the unit mass at the empty configuration is the only invariant probability of the contact process if \(\lambda \in \left ]0,\lambda _{c} \right [\), while for \(\lambda >\lambda _{c}\) there is another extremal (and translational invariant) probability \(\nu _{\lambda}\) satisfying \(\rho (\lambda)=\{\eta \in \{0,1\}^{\mathbf Z};\;\eta (x)=1\}>0\). The function \(\rho \) is continuous and \(\rho (\lambda _{c})=0\); the author finds a lower bound for \(\rho \) on \(\left ]\lambda _{c},+\infty \right [\), namely, he shows that \[ \rho (\lambda)\geq \left (1-\frac {\lambda}{\lambda _{c}} \right) \frac {2\lambda ^{2}+3\lambda +2} {(\lambda ^{2}+\lambda)(1+\lambda _{c})}, \quad \lambda >\lambda _{c}. \] An application of this estimate to a model of epidemics is given.