The $\phi^4$ theory on a hypercubic lattice with $d$ space-time dimensions is "trivial" for $d\geq 4$, in the sense that it reduces to a free non-interacting theory in the limit that the lattice spacing $a$ goes to zero. A rigorous proof for $d=4$ has been published only recently (2021), by Aizenman and Duminil-Copin in <A HREF="https://arxiv.org/abs/1912.07973">Marginal triviality of the scaling limits of critical 4D Ising and $\phi^4$ models</A>. A proof for $d>4$ was obtained earlier (1981), by <A HREF="https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.47.1">Aizenman</A>.

For finite $a$ interaction terms persist, and are relevant on energy scales below $1/a$, see <A HREF="https://doi.org/10.1016/0550-3213(87)90177-5">Lüscher and Weisz</A> (1987).