Séminaire de Mathématique

# Translation Invariant Gibbs Measures and Continuity for $\phi_d^4$ via Random Tangled Currents

## by Romain Panis (University of Geneva)

Europe/Paris
Amphithéâtre Léon Motchane (IHES)

### Amphithéâtre Léon Motchane

#### IHES

Le Bois Marie 35, route de Chartres 91440 Bures-sur-Yvette
Description

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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Organized by

Thierry Bodineau, Pieter Lammers, Yilin Wang

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