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 continuum limit. This has been rigorously proven for $d>4$, with numerical evidence for $d=4$. For small but nonzero lattice spacing $a$ at energies well below the cutoff mass $1/a$, the theory effectively behaves like a continuum theory with particle interactions.

See <A HREF="http://www.scholarpedia.org/article/Triviality_of_four_dimensional_phi%5E4_theory_on_the_lattice">Triviality of four dimensional $\phi^4$ theory on the lattice</A> for references to the literature.