Critical two-point function for the \(\varphi^4\) model in dimensions \(d>4\)

The (lattice) \(\varphi^4\) model is a scalar field-theoretical model that exhibits a phase transition. It is believed to be in the same universality class as the Ising model. In fact, we can construct the \(\varphi^4\) model as the \(N\to\infty\) limit of the sum of \(N\) Ising systems (with the right scaling of spin-spin couplings). Using this Griffiths-Simon construction and applying the lace expansion for the Ising model, we can prove mean-field asymptotic behavior for the critical \(\varphi^4\) two-point function. In this talk, I will explain the key points of the proof, and discuss possible extensions of the results to the power-law coupling case.