The on-site potential of the <Phi-four lattice model> is
$$
V(s)=\frac g4s^4+\frac\nu2s^2,
\qquad V''(s)=3gs^2+\nu\geq\nu>0.
$$
In the <random-walk representation of a lattice-field covariance>, $\langle\phi_x\phi_y\rangle$ 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, so
$$
0\leq\langle\phi_x\phi_y\rangle
\leq(-\Delta+\nu)^{-1}(x,y).
$$
The massive lattice Green function has the killed-walk expansion
$$
(-\Delta+\nu)^{-1}(x,y)
=\sum_{k\geq0}\frac1{2d+\nu}
\left(\frac{2d}{2d+\nu}\right)^k
\mathbb P_x(S_k=y),
$$
with 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
$$
|\langle\phi_x\phi_y\rangle|\leq Ce^{-c|x-y|_1}.
$$

Solution

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

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.


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