Metastability for a Non-reversible Dynamics: The Evolution of the Condensate in Totally Asymmetric Zero Range Processes

It has been observed (Evans in Braz J Phys 30:42-57, 2000; Jeon et al. in Ann Probab 28:1162-1194, 2000) that some zero-range processes exhibit condensation, a macroscopic fraction of particles concentrates on one single site. We examined in (Beltran and Landim in Probab Theory Relat Fields 152:781-807, 2012) the asymptotic evolution of the condensate in the case where the dynamics is reversible, the number of sites is fixed, and the total number of particles diverges. We proved in that paper that in an appropriate time-scale the condensate evolves according to a symmetric random walk whose transition rates are proportional to the capacities of the underlying random walk. In this article, we extend this result to the condensing totally asymmetric zero-range process, a non-reversible dynamics.