Ginibre interacting Brownian motion in infinite dimensions is sub-diffusive

TL;DR

This paper proves that in a two-dimensional infinite particle system with logarithmic interactions, tagged particles exhibit sub-diffusive behavior, highlighting the influence of geometric rigidity on stochastic dynamics.

Contribution

It demonstrates sub-diffusivity of tagged particles in Ginibre interacting Brownian motion, contrasting with diffusive behavior in higher dimensions.

Findings

Tagged particles are sub-diffusive in the Ginibre interacting Brownian motion.

The dynamics are reversible with respect to the Ginibre random point field.

Geometric rigidity of the particle system affects its dynamical properties.

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.