Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 209 / 4 / b / Solution

The on-site potential of the Phi-four lattice model isIn the random-walk representation of a lattice-field covariance, $⟨{\varphi }_x{\varphi }_y⟩$ is a Green function for a nearest-neighbour walk with a nonnegative environment-dependent killing rate bounded below by a positive constant depending on $\nu$. Dropping the quartic contribution can only decrease that killing, soThe massive lattice Green function has the killed-walk expansionwith normalization adjusted to the chosen Laplacian. Reaching $y$ requires at least $|x-y{|}_1$ steps, while the geometric survival factor is strictly below one. Summing the tail gives constants $C,c>0$ such that