Survival probability and field theory in systems with absorbing states

An important quantity in the analysis of systems with absorbing states is the survival probability $P_s\mathrm{}\left(t\right)\mathrm{}$, the probability that an initial localized seed of particles has not completely disappeared after time $t$. At the transition into the absorbing phase, this probability scales for large $t$ like $t^{-\delta }$. It is not at all obvious how to compute $\delta$ in continuous field theories, where $P_s\mathrm{}\left(t\right)\mathrm{}$ is strictly unity for all finite $t$. We propose here an interpretation for $\delta$ in field theory and devise a practical method to determine it analytically. The method is applied to field theories representing absorbing-state systems in several distinct universality classes. Scaling relations are systematically derived and the known exact $\delta$ value is obtained for the voter model universality class.