Probability and Mathematical Physics Seminar

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Probability and the City seminar

Speaker: Volodymyr Riabov (ISTA) and Pawel Duch (EPFL)

Location: Warren Weaver Hall 1302

Date: Friday, September 19, 2025, 11:10 a.m.

Synopsis:

Volodymyr Riabov (ISTA):

TBA

Pawel Duch (EPFL):

Ergodicity of infinite volume $\Phi^4_3$ model at high temperature

The dynamical $\Phi^4_3$ model is a stochastic partial differential equation that arises in quantum field theory and statistical physics. Owing to the singular nature of the driving noise and the presence of a nonlinear term, the equation is inherently ill-posed. Nevertheless, it can be given a rigorous meaning, for example, through the framework of regularity structures. On compact domains, standard arguments show that any solution converges to the equilibrium state described by the unique invariant
measure. Extending this result to infinite volume is highly nontrivial: even for the lattice version of the model, uniqueness holds only in the high-temperature regime, whereas at low temperatures multiple phases coexist.

We prove that, when the mass is sufficiently large or the coupling constant sufficiently small (that is, in the high-temperature regime), all solutions of the dynamical $\Phi^4_3$ model in infinite volume converge exponentially fast to the unique stationary solution, uniformly over all initial conditions. In particular, this result implies that the invariant measure of the dynamic is unique, exhibits exponential decay of correlations and is invariant under translations, rotations and reflections.

Joint work arXiv:2508.07776 with Martin Hairer, Jaeyun Yi and Wenhao Zhao.