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 the power law $r(q,t)\sim t^{-\theta \left(q\right)}$ with a nontrivial dependence of the exponent $\theta \left(q\right)$ on $q$ and on space dimension. By mapping the problem on an exactly soluble one-species coagulation model ( $A+A\to A$), we obtain the exact expression of $\theta \left(q\right)$ in dimension one.