# Condensation of SIP particles and sticky Brownian motion

## TL;DR

This paper investigates the condensation behavior of symmetric inclusion process (SIP) particles, deriving explicit variance scaling and demonstrating convergence to sticky Brownian motion, thereby advancing understanding of coarsening dynamics in particle systems.

## Contribution

It provides the first explicit variance scaling in the condensation regime of SIP and proves convergence to sticky Brownian motion using Mosco convergence of Dirichlet forms.

## Key findings

- Explicit variance scaling for SIP in condensation regime
- Convergence of SIP particle differences to sticky Brownian motion
- Probabilistic analysis of particles being together over time

## Abstract

We study the symmetric inclusion process (SIP) in the condensation regime. We obtain an explicit scaling for the variance of the density field in this regime, when initially started from a homogeneous product measure. This provides relevant new information on the coarsening dynamics of condensing interacting particle systems on the infinite lattice. We obtain our result by proving convergence to sticky Brownian motion for the difference of positions of two SIP particles in the sense of Mosco convergence of Dirichlet forms. Our approach implies the convergence of the probabilities of two SIP particles to be together at time $t$. This, combined with self-duality, allows us to obtain the explicit scaling for the variance of the fluctuation field.