Limit Theorems for Tagged Particles

#### Pablo A. Ferrari

1996, v.2, №1, 17-40

ABSTRACT

We review old and new results about the limiting behaviour of a tagged particle in different interacting particle systems: (a) independent particles with no mass in one dimension with continuous paths like Brownian motions and ideal gases; (b) reversible processes on $R^d$ or $Z^d$ like interacting Brownian motions and Kawasaki dynamics; (c) simple exclusion processes; (d) zero range processes; (e) conservative nearest particle processes including the Hammersley process. In each case we consider as initial distribution the invariant distribution for the process as seen from the tagged particle. We review two kinds of limiting behaviour: the law of large numbers and the invariance principle. Denoting by $X(t)$ the position of the tagged particle, the law of large numbers says that as $t\to\infty$, $X(t)/t$ converges almost surely to a constant. The invariance principle means that when conveniently centered and rescaled, the process converges to Brownian motion. When the initial distribution of the particles is not invariant, a weak form of the law of large numbers has been obtained in some cases.

Keywords: simple exclusion process,zero range process,tagged particles,central limit theorem,invariance principle,law of large numbers