Phi-four lattice model (${\varphi }^4$)

The phi-four lattice model is a real scalar lattice field with nearest-neighbour gradient energy and an on-site potential $g{\varphi }^4/4+\nu {\varphi }^2/2$.

The Helffer-Sjöstrand random-walk representation expresses a lattice-field covariance as the Green function of a walk in an environment determined by the Hessian of the field potential. A uniformly positive Hessian supplies killing and exponential covariance decay.