Field-theoretic approach to metastability in the contact process

A “quantum” field-theoretic formulation of the dynamics of the contact process on a regular graph of degree z is introduced. A perturbative calculation in powers of $1/z$ of the effective potential for the density of particles $\phi \mathrm{}\left(t\right)\mathrm{}$ and an instantonic field $\psi \mathrm{}\left(t\right)\mathrm{}$ emerging from the formalism is performed. Corrections to the mean-field distribution of densities of particles in the out-of-equilibrium stationary state are derived in powers of $1/z.$ Results for typical (e.g., average density) and rare fluctuation (e.g. lifetime of the metastable state) properties are in very good agreement with numerical simulations carried out on D-dimensional hypercubic $\mathrm{}(z=2D)\mathrm{}$ and Cayley lattices.