Exact results for curvature-driven coarsening in two dimensions

We consider the statistics of the areas enclosed by domain boundaries (`hulls’) during the curvature-driven coarsening dynamics of a two-dimensional nonconserved scalar field from a disordered initial state. We show that the number of hulls per unit area that enclose an area greater than $A$ has, for large time $t$, the scaling form $N_h(A,t)=2c\mathrm{/}(A+\lambda t)$, demonstrating the validity of dynamical scaling in this system, where $c=1\mathrm{/}8\pi \sqrt{3}$ is a universal constant. Domain areas (regions of aligned spins) have a similar distribution up to very large values of $A\mathrm{/}\lambda t$. Identical forms are obtained for coarsening from a critical initial state, but with $c$ replaced by $c\mathrm{/}2$.