Exact results for two-dimensional coarsening

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, $n_h(A,t)\,dA$, with enclosed area in the range (A, A+dA), is described, for large time t, by the scaling form $n_h(A,t) = 2c_h/(A+\lambda_h t)^2$, demonstrating the validity of dynamical scaling in this system. Here $c_h = 1/8\pi\sqrt{3}$ is a universal constant associated with the enclosed area distribution of percolation hulls at the percolation threshold, and $\lambda_h$ is a material parameter. The distribution of domain areas, $n_d(A,t)$, is apparently very similar to that of hull areas up to very large values of $A/\lambda_h t$. Identical forms are obtained for coarsening from a critical initial state, but with ch replaced by $c_h/2$. The similarity of the two distributions (of areas enclosed by hulls, and of domain areas) is accounted for by the smallness of ch. By applying a `mean-field' type of approximation we obtain the form $n_d(A,t) \simeq 2c_d[\lambda_d(t+t_0)]^{\tau-2}/[A +\lambda_d(t+t_0)]^\tau$, where t0 is a microscopic timescale and $\tau = 187/91 \simeq 2.055$, for a disordered initial state, and a similar result for a critical initial state but with $c_d \to c_d/2$ and $\tau \to \tau_c =379/187 \simeq 2.027$. We also find that $c_d = c_h + O(c_h^2)$ and $\lambda_d = \lambda_h(1+O(c_h))$. These predictions are checked by extensive numerical simulations and found to be in good agreement with the data