Marginal Triviality Of The Scaling Limits Of Critical 4d Ising And Phi(4)(4) Models

We prove that the scaling limits of spin fluctuations in four-dimensional Isingtype models with nearest-neighbor ferromagnetic interaction at or near the critical point are Gaussian. A similar statement is proven for the lambda phi(4) fields over R-4 with a lattice ultraviolet cutoff, in the limit of infinite volume and vanishing lattice spacing. The proofs are enabled by the models random current representation, in which the correlation functions deviation from Wicks law is expressed in terms of intersection probabilities of random currents with sources at distances which are large on the models lattice scale. Guided by the analogy with random walk intersection amplitudes, the analysis focuses on the improvement of the so-called tree diagram bound by a logarithmic correction term, which is derived here through multi-scale analysis.