Application of the lace expansion to the $\varphi^4$ model

Abstract

Using the Griffiths-Simon construction of the $\varphi^4$ model and the lace expansion for the Ising model, we prove that, if the strength $\lambda\ge0$ of nonlinearity is sufficiently small for a large class of short-range models in dimensions $d>4$, then the critical $\varphi^4$ two-point function $\langle\varphi_o\varphi_x\rangle_{\mu_c}$ is asymptotically $|x|^{2-d}$ times a model-dependent constant, and the critical point is estimated as $\mu_c=\mathscr{\hat J}-\frac\lambda2\langle\varphi_o^2\rangle_{\mu_c}+O(\lambda^2)$, where $\mathscr{\hat J}$ is the massless point for the Gaussian model.