# The Widom-Rowlinson model: Mesoscopic fluctuations for the critical droplet

## TL;DR

This paper rigorously analyzes the mesoscopic surface fluctuations of the critical droplet in a two-dimensional Widom-Rowlinson particle system at low temperature, advancing understanding of phase separation in continuum models.

## Contribution

It provides the first detailed mathematical analysis of surface fluctuations in a continuum particle system undergoing condensation.

## Key findings

- Surface fluctuations are close to a deterministic disk shape.
- Results establish a foundation for non-equilibrium Widom-Rowlinson models.
- Analysis aids in refining the Arrhenius formula for phase transition times.

## Abstract

We study the critical droplet for a close-to-equilibrium Widom-Rowlinson model of interacting particles, represented by disks of radius $1$, in the two-dimensional plane at low temperature. The critical droplet is the set of macroscopic states that correspond to saddle points for the passage from a low-density supersaturated vapour to a stable high-density liquid. We analyse the mesoscopic fluctuations of the surface of the critical droplet, which turns out to be the set of particle configurations that are close to a disk of a certain deterministic radius. Our results represent the first detailed rigorous analysis of the surface fluctuations of a continuum interacting particle system exhibiting condensation and, as such, constitute a fundamental step in the study of phase separation from the perspective of stochastic geometry. At the same time, our results serve as a basis for the study of a non-equilibrium version of the Widom-Rowlinson model, to be analysed elsewhere, where they lead to a correction term in the Arrhenius formula for the average vapour-liquid crossover time.