We obtain θ_p(q) = 2θ_s(q) for one-dimensional q-state ferromagnetic Potts models evolving under parallel dynamics at zero temperature from an initially disordered state, where θ_p(q) is the persistence exponent for parallel dynamics and θ_s(q) = -{1/8}+ \frac{2}{π^2}[cos^{-1}{(2-q)/q\sqrt{2}}]^2 [PRL, {\bf 75}, 751, (1995)], the persistence exponent under serial dynamics. This result is a consequence of an exact, albeit non-trivial, mapping of the evolution of configurations of Potts spins under parallel dynamics to the dynamics of two decoupled reaction diffusion systems.
We study a d-dimensional system of diffusing particles that on contact either annihilate with probability $1/(q\ensuremath{-}1)$ or coagulate with probability $(q\ensuremath{-}2)/(q\ensuremath{-}1).$ In one dimension, the system models the zero-temperature Glauber dynamics of domain walls in the q-state Potts model. We calculate
P(m,t), the probability that a randomly chosen lattice site contains a particle whose ancestors have undergone exactly $(m\ensuremath{-}1)$ coagulations. Using perturbative renormalization group analysis for
d<2, we show that, if the number of coagulations m is much less than the typical number
M(t), then $\overline{P}(m,t)\ensuremath{\sim}{m}^{\ensuremath{\zeta}/d}{t}^{\ensuremath{-}\ensuremath{\theta}},$ with $\ensuremath{\theta}=dQ+Q(Q\ensuremath{-}1/2)\ensuremath{\epsilon}+O({\ensuremath{\epsilon}}^{2}),$ $\ensuremath{\zeta}=(2Q\ensuremath{-}1)\ensuremath{\epsilon}+(2Q\ensuremath{-}1)(Q\ensuremath{-}1)(1/2+AQ){\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3}),$ where $Q=(q\ensuremath{-}1)/q,$ $\ensuremath{\epsilon}=2\ensuremath{-}d$ and $A=\ensuremath{-}0.006\dots{}.$
M(t) is shown to scale as $M(t)\ensuremath{\sim}{t}^{d/2\ensuremath{-}\ensuremath{\delta}},$ where $\ensuremath{\delta}=d(1\ensuremath{-}Q)+(Q\ensuremath{-}1)(Q\ensuremath{-}1/2)\ensuremath{\epsilon}+O({\ensuremath{\epsilon}}^{2}).$ In two dimensions, we show that $\overline{P}(m,t)\ensuremath{\sim}\mathrm{ln}{(t)}^{Q(3\ensuremath{-}2Q)}\mathrm{ln}{(m)}^{(2Q\ensuremath{-}{1)}^{2}}{t}^{\ensuremath{-}2Q}$ for $m\ensuremath{\ll}{t}^{2Q\ensuremath{-}1}.$ We also derive an exact nonperturbative relation between the exponents: namely $\ensuremath{\delta}(Q)=\ensuremath{\theta}(1\ensuremath{-}Q).$ The one-dimensional results corresponding to $\ensuremath{\epsilon}=1$ are compared with results from Monte Carlo simulations.
For the Glauber dynamics of the one-dimensional q-state Potts model at zero temperature, the fraction of spins which have never flipped up to time t decreases as t - theta (q ) where the exponent theta (q) varies continuously with q. The calculation of the q dependence of this exponent can be reduced to the study of a simple reaction-diffusion model. Using finite-size scaling, this allows one to obtain accurate estimates of B(q) for all values of q. One can also show that for 1=1+ epsilon , the expansion of theta (y) is given by theta (1)=(2/ pi ) epsilon +O( epsilon 2 ).