Probability and analysis informal seminar
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 $\phi^4_d$ model on $Z^d$. We prove that the set of translation invariant Gibbs measures for the $\phi^4_d$ model on $Z^d$ 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 $\phi^4_d$ model on $$Z^d$ vanishes at criticality for d>=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 $\phi^4_d$ model called the random tangled current representation.
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Thierry Bodineau, Pieter Lammers, Yilin Wang
