Marginal triviality of the scaling limits of 4D critical Ising and Phi^4 models

The talk will present the recent proof that in four dimensions the spin fluctuations of Ising-type models at their critical points are Gaussian in their scaling limits (infinite volume, vanishing lattice spacing). Similar statement is proven for the scaling limits of more general PHI^4 fields constructed through a lattice cutoff. The proofs are facilitated by the systems’ random current representation, in which the deviation from Wick's law are expressed in terms of intersection probabilities of random currents with prescribed sources. This approach previously yielded such statements for D>4. Their recent extension to the marginal dimension was enabled by a multiscale analysis of the critical clusters’ intersections.

Joint work with Hugo Duminil-Copin.