Convergence of the Condensing Symmetric Inclusion Process on the Torus in the Thermodynamical Limit to Coalescing Brownian Motions

TL;DR

This paper proves that, in the thermodynamical limit, condensate positions in a symmetric inclusion process on a torus behave like coalescing Brownian motions, merging upon contact.

Contribution

It provides a rigorous derivation of coalescing Brownian motions as the limit of condensate dynamics in an interacting particle system with condensation.

Findings

Condensate positions converge to coalescing Brownian motions.

Condensates perform diffusive motion until merging.

Coalescence times are negligible compared to movement times.

Abstract

We investigate the saturation regime of the condensing symmetric inclusion process on the discrete one-dimensional torus in the thermodynamical limit. In this regime, the total mass concentrates on a finite number of sites, forming condensates. Our main result establishes that, under appropriate scaling, the positions of the condensates converge to a system of coalescing Brownian motions on the continuum torus. In particular, condensates perform diffusive motion until they meet, at which point they merge and their masses coagulate. This provides a rigorous derivation of a macroscopic coalescing diffusion from an underlying interacting particle system with condensation. The main technical difficulty arises from the complicated coalescence mechanism of two condensates of particles, whose trajectories are very difficult to track completely. The key idea is to control the coalescing time instead and prove that it is negligible compared to the time-scale of condensate movement. By combining this with precise estimates of movements without coalescence, we can prove its convergence to coalescing Brownian motions.