we show that activated processes restore, after a characteristic time $\tau _p^{cross}(T_f)$, the Ising behavior also in the cases with $p>p_c$.

This paper is organized as follows: In Sec. \[model\] we introduce the model and define the observable quantities that will be considered. In Sec. \[num\] we present the outcome of numerical simulations of the model with different $p$. In particular, quenches to $T_f=0$ or to $T_f>0$ will be discussed in Secs. \[Tzer\] and \[Tnzer\], respectively. Here we enlighten the crossover between the Ising and the XY universality class and provide an argument explaining its microscopic origin. A summary and the conclusions are contained in Sec. \[concl\].

Model and observables {#model}
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The $p$-state clock model in one dimension is defined by the Hamiltonian H\[\]=-J\_[i=1]{}\^[N]{}\_i \_[i+1]{}= -J\_[i=1]{}\^[N]{}cos(\_i-\_[i+1]{}), \[hamiltonian\] where $\vec \sigma _i$ is a two-components unit vector spin pointing along one of the directions \_i= n\_i, \[theta\] with $n_i \in \{1,2,...,p\}$, $i=1,...,{\cal N}$ are the sites of the lattice and we assume periodic boundary conditions $\theta _{{\cal N}+1}=\theta _1$. This spin system is equivalent to the Ising model for $p=2$ and to the XY model for $p\to \infty $. In $d=1$ the system is ergodic except at $T=0$. At any 