for $p=\infty $. We study how the model with generic $p$ interpolates between these limiting cases which, as discussed above, behave in a radically different way. In doing that, we uncover the existence of a critical value $p_c=4$, where the properties of the dynamics change abruptly. For $p\le p_c$ the dynamics at $T_f=0$ is characterized by Brownian motion and annihilation of interfaces between domains, as in the Ising model. One has dynamical scaling with the same dynamical exponents and, interestingly, the same scaling functions of the Ising model. For $p>p_c$, instead, the dynamics is dominated by a texture mechanism analogous to the case with $p=\infty $, and dynamical scaling is violated.

In $d=1$ there is no possibility of ergodicity breaking except at $T=0$. At any finite temperature the equilibrium state is disordered with a vanishing magnetization and a coherence length $\xi (T)$ that diverges in the $T\to 0$ limit. If the system is quenched to a sufficiently low temperature one has a coarsening phenomenon in a pre-asymptotic transient until the growing length associated to the development of order becomes comparable with $\xi (T_f)$. Since $\xi (T_f)$ diverges as $T_f\to 0$ the phase-ordering stage can be rather long. In this regime 