Wed, 18 Nov 2026
11:00 -
13:00
L4
Uniqueness of the infinite volume $\Phi^4_3$ measure up to critical temperature
Jacopo Peroni
(University of Muenster)
Abstract
The $\Phi^4_3$ measure in infinite volume is obtained as a limit of the measures $\nu_L$ on the tori $\mathbb{T}^3_L$ as $L\to\infty$. Tightness of the $\nu_L$ is known and gives limit points, but it says nothing about their uniqueness, nor does it rule out invariant measures of the infinite-volume Langevin dynamics unrelated to the $\nu_L$. At weak coupling, uniqueness was proved by Duch, Hairer, Yi and Zhao through a contraction property of the dynamics. In this talk I will show that, at every coupling $\lambda>0$, the $\Phi^4_3$ dynamics on the full space has exactly one invariant measure for every mass $\mu>\mu_c(\lambda)$, that is, in the whole high-temperature phase, where the susceptibility of the lattice $\varphi^4_3$ measures is bounded uniformly in the mesh and in the volume. In addition, the torus measures $\nu_L$ converge to it almost exponentially fast in $L$, and the dynamics converges to it exponentially fast in time, uniformly in the initial condition. The proof follows the method of Holley and Stroock, which Bauerschmidt, Dagallier and Weber used in two dimensions. The main new ingredient is the entropy bound in three dimensions, which we obtain from a dimension-free log-Harnack inequality. Joint work with Hendrik Weber.
