for each simulation. The statistical errors in the data reported in the figures are always smaller than the dimension of the symbols or the thickness of the lines.

Quenches to $T_f=0$. {#Tzer}
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Let us start with quenches to $T_f=0$, by illustrating the behavior of the characteristic length $L_G(t)$ defined in Eq. (\[halfheight\]). In Fig. \[figlengthT0\] this quantity is plotted against $t^{1/2}$ (left panel) or against $t^{1/4}$ (right panel), for several values of $p$ ranging from $p=2$ to $p=\infty $.

This figure shows that $L_G(t)$ has an asymptotic power law growth, as in Eq. (\[ldit\]), for every value of $p$. However, the dynamic exponent $z$ radically changes going from $p\le p_c$, where one has values very well consistent with $z=2$ (best fits yield $1/z=0.49\pm 0.01$ for $p=2,3,4$), to $p>p_c$ where $z=4$ is found with good accuracy (we find $1/z=0.27\pm 0.01, 0.27\pm 0.01, 0.25\pm 0.01$ for $p=5,6,10$). We recall that these are the values found in the Ising model and in the XY model. The behavior of $L_G(t)$, then, indicates a crossover from Ising to XY behavior upon crossing $p_c=4$. We will see in the following that this is confirmed by the analysis of other dynamical quantities. Before doing that, however, 