$L_G(t)$ for $p=6$ and different values of $T_f$. Here one observes initially the same behavior as for $T_f=0$, namely $L_G(t)\propto t^{1/4}$, i.e. a straight line in the plot of $L_G(t)$ against $t^{1/4}$ (right panel). For larger times there is a crossover to the Ising behavior $L_G(t)\simeq t^{1/2}$, namely a straight line in the plot of $L_G(t)$ versus $t^{1/2}$ (left panel). Although the crossover is a quite smooth phenomenon, as can be seen in Fig. \[figlengthT2\], $\tau _p^{cross}(T_f)$ given by Eq. (\[tcross\]), represented by thick segments across the lines, turns out to be of the correct order of magnitude for all the temperatures considered.

In Fig. \[figlengthT1\] we plot $L_G(t)$ for $T_f=0.1$ and different values of $p$. One observes the same pattern of behavior of Fig. \[figlengthT2\] with a crossover from a power law growth with $z=4$ to one with $z=2$. The crossover time (\[tcross\]) grows with $p$, as expected.

Conclusions {#concl}
===========

In this paper we have studied the phase-ordering kinetics of the one dimensional $p$-state clock model. We have shown the existence of a critical value $p_c=4$ separating two radically different dynamical behaviors. For $p\le p_c$ the dynamics is in all respects analogous to that of the Ising model 