  We study the critical droplet for a close-to-equilibrium Widom-Rowlinson
model of interacting particles in the two-dimensional continuum at low
temperatures. The critical droplet is the set of macroscopic states that
correspond to saddle points for the passage from a low-density vapour to a
high-density liquid. We first show that the critical droplet is close to a disc
of a certain deterministic radius. After that we analyse the mesoscopic
fluctuations of the surface of the critical droplet. These are built on
microscopic fluctuations of the particles in the boundary layer, and provide a
rigorous foundation for what in physics literature is known as capillary waves.
  Our results represent the first detailed analysis of the surface fluctuations
down to mesoscopic and microscopic precision. As such they constitute a
fundamental step in the study of phase separation in continuum interacting
particle systems from the perspective of stochastic geometry. At the same time,
they 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.
