This seminar will be presented in hybrid mode.  The speaker will deliver his talk in person.

Title: Translation invariant Gibbs measures and continuity for φ^4_d via random tangled currents

Abstract: In this talk I will present recent results obtained in joint work with Trishen Gunaratnam, Christoforos Panagiotis and Franco Severo concerning the study of Gibbs measures of the lattice $\varphi^4_d$ model on $\mathbb Z^d$. We prove that the set of translation invariant Gibbs measures for the $\varphi^4_d$ model on $\mathbb Z^d$ has at most two extremal measures at all temperature. We also  give a sufficient condition to ensure that the set of all Gibbs measures is a singleton. As an application, we show that the spontaneous magnetisation of the nearest-neighbour $\varphi^4_d$ model on $\mathbb Z^d$ vanishes at criticality for $d\geq 3$. The analogous results were established for the Ising model in the seminal works of Aizenman, Duminil-Copin, and Sidoravicius (Comm.  Math. Phys., 2015), and Raoufi (Ann. Prob., 2020) using the so-called random current representation introduced by Aizenman (Comm.\ Math.\ Phys., 1982). Our proof relies on a new corresponding stochastic geometric representation for the $\varphi^4_d$ model called the random tangled current representation.

The talk will be followed by refreshments in the Huxley Common Room at 5pm.