Ginibre interacting Brownian motion in infinite dimensions is sub-diffusive

Abstract

We prove that the tagged particles of infinitely many Brownian particles in $ \Rtwo $ interacting via a logarithmic (two-dimensional Coulomb) potential with inverse temperature $ \beta = 2 $ are sub-diffusive. The associated delabeled diffusion is reversible with respect to the Ginibre random point field, and the dynamics are thus referred to as the Ginibre interacting Brownian motion. % If the interacting Brownian particles have interaction potential $ \Psi $ of Ruelle class and the total system starts in a translation-invariant equilibrium state, then the tagged particles are always diffusive if the dimension $ \dd $ of the space $ \mathbb{R}^{\dd } $ is greater than or equal to two. That is, the tagged particles are always non-degenerate under diffusive scaling. Our result is, therefore, contrary to known results. The Ginibre random point field has various levels of geometric rigidity. Our results reveal that the geometric property of infinite particle systems affects the dynamical property of the associated stochastic dynamics.