Fluid limit for the coarsening phase of the condensing zero-range process

TL;DR

This paper establishes a fluid limit for the coarsening phase of the condensing zero-range process, describing the macroscopic evolution of the system via piecewise linear trajectories in a finite-dimensional simplex.

Contribution

It introduces a novel fluid limit characterization for the coarsening phase, linking the process to a trace process of an underlying random walk and identifying absorbing configurations.

Findings

The process converges to a piecewise linear trajectory in the simplex.

Absorbing configurations are explicitly characterized.

Finite-time absorption in the fluid limit is proven.

Abstract

We prove a fluid limit for the coarsening phase of the condensing zero-range process on a finite number of sites. When time and occupation per site are linearly rescaled by the total number of particles, the evolution of the process is described by a piecewise linear trajectory in the simplex indexed by the sites. The linear coefficients are determined by the trace process of the underlying random walk on the subset of non-empty sites, and the trajectory reaches an absorbing configuration in finite time. A boundary of the simplex is called absorbing for the fluid limit if a trajectory started at a configuration in the boundary remains in it for all times. We identify the set of absorbing configurations and characterize the absorbing boundaries.