Condensation in the inclusion process and related models

TL;DR

This paper investigates condensation phenomena in various particle systems related to the inclusion process, revealing conditions under which particles concentrate on specific sites, including asymmetric, independent, and continuous models, with detailed asymptotic behaviors.

Contribution

It extends condensation results to asymmetric inclusion processes, independent variables with different tails, and the Brownian energy process, providing a unified understanding of condensation across models.

Findings

Condensation occurs on the rightmost site in asymmetric systems.

Heavy-tailed distributions determine the site of condensation.

Condensation persists in the limit of vanishing diffusion.

Abstract

We study condensation in several particle systems related to the inclusion process. For an asymmetric one-dimensional version with closed boundary conditions and drift to the right, we show that all but a finite number of particles condense on the right-most site. This is extended to a general result for independent random variables with different tails, where condensation occurs for the index (site) with the heaviest tail, generalizing also previous results for zero-range processes. For inclusion processes with homogeneous stationary measures we establish condensation in the limit of vanishing diffusion strength in the dynamics, and give several details about how the limit is approached for finite and infinite systems. Finally, we consider a continuous model dual to the inclusion process, the so-called Brownian energy process, and prove similar condensation results.