Seminar:Non-triviality of the radial $\Phi^4_d$-model for $d \ge 4$.

Seminar| Institute of Mathematical Sciences

Time:Thursday, April 24th, 2025,16:00-17:00

Abstract:The $\Phi^4_d$-measure is the central object in constructive quantum field theory. Its construction for $d = 2, 3$ is one of the early achievements in the field. For $d \ge 4$, the $\Phi^4_d$-measure is known to be trivial with a recent breakthrough work for the 4-$d$ case by Aizenman and Duminil-Copin (2021).

In this talk, by imposing the radial assumption, I will present the construction of the $\Phi^4_d$-measure for $d \ge 4$ on the unit ball by considering the limit of the $\Phi^4_d$-measures on punctured balls. In order to control the singularity at the origin, we exploit the Markov property of the Gaussian free field over disjoint sets and sharp Green's function estimates on a punctured ball. We also show that the resulting $\Phi^4_d$-measure has a sub-Gaussian tail and that it is singular with respect to the base Gaussian free field.

This is based on a joint work with Tadahiro Oh (the University of Edinburgh and Beijing Institute of Technology), Leonardo Tolomeo (the University of Edinburgh), and Nikolay Tzvetkov (ENS Lyon).