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

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.

Figures (12)

• Figure 1: A particle configuration $\gamma\in\Gamma$ (with a penetrable unit disk $B(x)$ around each particle $x\in\gamma$) and its halo $h(\gamma)$ (determining the energy of $\gamma$).
• Figure 2: Graph of the coexistence line $\beta \mapsto z_t(\beta)$.
• Figure 3: Picture of $R \mapsto \Phi_\kappa(R)$ for fixed $\kappa\in(1,\infty)$ and $\kappa \mapsto R_{\mathrm{c}}(\kappa)$.
• Figure 4: A picture of the critical droplet in the metastable regime \ref{['metaregalt']}. The critical droplet is given by a cluster of particles close to $B_{R_{\mathrm{c}}(\kappa)}$, a disk of radius $R_{\mathrm{c}}(\kappa)$. It has a random boundary that fluctuates within a narrow annulus whose width shrinks to zero as $\beta\to\infty$. In the course of the paper it will become clear that the number of unit disks lying fully inside $B_{R_{\mathrm{c}}(\kappa)}$ is $\kappa\beta$ (modulo a constant), while the number of unit disks that touch the boundary $\partial B_{R_{\mathrm{c}}(\kappa)}$ is $\frac{\kappa^{2/3}}{\kappa-1}\beta^{1/3}$ (modulo a constant).
• Figure 5: The set $\boldsymbol{z}(\gamma)$ consisting of the boundary points of the configuration $\gamma$ from Fig. \ref{['fig:critdrop']} is shown on the left. The thick line is the boundary $\partial h(\gamma)$ of the halo $h(\gamma)$. On the right, the boundary layer $S(\boldsymbol{z})\setminus S(\boldsymbol{z})^-$ is shown in blue with the outer boundary $\partial h(\gamma)$ the same as on the left picture and the inner boundary $\partial h(\gamma)^-$ consisting of circular arcs of unit disks in pale red with centres at the cusps of the outer boundary.

Theorems & Definitions (82)

• Theorem 1.1: Large deviation principle for the halo shape
• Theorem 1.2: Minimisers of the rate function for the halo shape
• Theorem 1.3: Large deviation principle for the halo volume
• Theorem 1.4: Moderate deviation bounds
• Conjecture 1.5: Moderate deviations
• Remark 1.6: Three conditions
• Remark 1.7: Metastability
• Remark 1.8: Broader context
• Lemma 2.1
• Lemma 2.2