Dynamics for exclusion processes with random environments of particle jump rates

Abstract

We consider a continuous-time particle system with exclusion interaction on the integer lattice. Each particle is assigned intrinsic left and right jump rates by an independent draw from a common random environment. We identify the decomposition of the system into maximal stable subsystems, which we call clouds. We show different qualitative behaviour for the cloud decomposition corresponding to different regimes for the random environment law that parallel the classical Sinai and Kesten--Kozlov--Spitzer regimes from one-dimensional random walk in random environment. Our main result in the Sinai regime is that the system is decomposed into an infinite number of finite clouds which all go to minus infinity with rapidly decreasing speed magnitudes, and we obtain explicit associated distributional limits. We analyse the model through the potential function associated with the random environment. In the Sinai regime, deep potential wells trap particles, producing stable clouds whose speeds are exponentially small in the well depth. Since successive wells are deeper, the resulting clouds are successively slower. Thus the localization mechanism of Sinai's random walk manifests itself here as a jamming phenomenon, in which a potential well traps not one walker but an entire block of mutually excluding particles.