MURST through PRIN-2004.

\[appendix1\]

For the 3-states clock model the correlation between two spins at a certain time $t$ can be written as G(r,t)=\_i \_j= \_[n,n’=1,3]{}P\_i(n,t)P\_[i,j]{}(n,tn’,t), where $r$ is the distance between $i$ and $j$. $n$, $\theta _i$ (and their relation) are defined in Eq. (\[theta\]), $P_i(n,t)$ is the probability to find the spin on site $i$ in the state $n$ at time $t$, and $P_{i,j}(n,t\mid n',t)$ is the conditional probability to find the state $n'$ on site $j$ provided that the state $n$ is found in $i$. Isolating the diagonal terms one has G(r,t) = \_[n=1,3]{}P\_i(n,t)P\_[i,j]{}(n,tn,t)- \_[n=1,3]{}P\_i(n,t)\_[n’n]{}P\_[i,j]{}(n,t n’,t), where we have used the value $\cos (\theta _i-\theta _j)=-1/2$ when $\theta _i\ne \theta _j$. Since $\sum _{n'\ne n}P_{i,j}(n,t\mid n',t)=1-P_{i,j}(n,t\mid n,t)$ one has G(r,t)= -\_[n=1,3]{}P\_i(n,t)+ \_[n=1,3]{}P\_i(n,t)P\_[i,j]{}(n,tn,t) =-+ \_[n=1,3]{}P\_i(n,t)P\_[i,j]{}(n,tn,t) \[appe3\]

Let us turn now to the Potts model where a generic spin on site $i$ can be found in the states labeled with $m_i=1,2,3$. Following Ref. [@Sire], we define an auxiliary field $\phi _i (n)$ such that $\phi _i (n)=1$ if $m_i=n$, where $n$ is a reference state, and $\phi _i (n)=0$ otherwise. The correlation of the auxiliary field is the [*single phase*]{} correlation function of the Potts model and can be 