Exact first-passage exponents of 1D domain growth: relation to a reaction diffusion model

In the zero temperature Glauber dynamics of the ferromagnetic Ising or $q$-state Potts model, the size of domains is known to grow like $t^{1/2}$. Recent simulations have shown that the fraction $r(q,t)$ of spins which have never flipped up to time $t$ decays like a power law $r(q,t) \sim t^{-\theta(q)}$ with a non-trivial dependence of the exponent $\theta(q)$ on $q$ and on space dimension. By mapping the problem on an exactly soluble one-species coagulation model ($A+A\rightarrow A$), we obtain the exact expression of $\theta(q)$ in dimension one.