Interacting particle systems in random environment

Many aspects of stochastic interacting systems are well understood when the underlying single-particle random walks are homogeneous in space. E.g., large scale limits (steady state fluctuations and hydrodynamic limits) of exclusion and zero-range processes based on simple (symmetric or asymmetric) random walks on Z^d have been much studied and are well understood by now. Much less is known when there is some randomness a priori encoded in the underlying spatial structure, like random jump rates or random local drift depending on the random environment.

In this project we plan to investigate interacting particle systems in divergence-free random drift fields. This means essentially that the graph edges are randomly oriented in such a way that the average local fluxes are nil, however the systems are far from being reversible. It is known that the random walk in this type of random environment is either diffusive or superdiffusive depending on whether a certain condition on the correlations of the random drift field are valid or not. We plan to investigate how this dichotomy is reflected on the large scale behaviour of the interacting particle systems based on these random walks. Extremal stationary measures will still be of product structure. This fact makes it realistically hopeful that hydrodynamic limits and steady state fluctuations can be investigated. However, the large scale limits can be very different from the usual ones. We will investigate the hydrodynamic limit and the steady state fluctuation fields of these kind of models.