On the Supercritical Phase of the $\phi^4$ Model

by Franco Severo (Institut Camille Jordan, Lyon 1)

Friday May 9, 2025, 2:00 PM → 4:00 PM Europe/Paris

Amphithéâtre Léon Motchane (IHES)

Probability and analysis informal seminar

The $\phi^4$ model is a real-valued spin system with quartic potential. This model has deep connections with the classical Ising model, and both are expected to belong to the same universality class. We construct a random cluster representation for $\phi^4$, analogous to that of the Ising model. For this percolation model, we prove that local uniqueness of macroscopic cluster holds throughout the supercritical phase. The corresponding result for the Ising model was proved by Bodineau (2005) and serves as the crucial ingredient in renormalization arguments used to study fine properties of the supercritical behaviour, such as surface order large deviations, the Wulff construction and exponential decay of truncated correlations. The unboundedness of spins in the $\phi^4$ model imposes considerable difficulties when compared with the Ising model. This is particularly the case when handling boundary conditions, which we do by relying on the recently constructed random current representation of the model.

Joint work with Trishen Gunaratnam, Christoforos Panagiotis and Romain Panis.

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