数理談話会

【講演概要】:
The lattice $\varphi^4$ model is a scalar field-theoretical model that is known to exhibit a phase transition. It is believed to be in the same universality class as Ising ferromagnets. In fact, we can construct the $\varphi^4$ model as the large-$N$ 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 dimensions higher than the upper-critical dimension. For the case of finite-range spin-spin couplings, in particular, the mean-field asymptotic behavior is Newtonian, and the upper-critical dimension is 4. I will explain the key ideas of the proof and discuss extension of the results to the case of power-law decaying spin-spin couplings.