Maes C. , Vande Velde K.
The fuzzy Potts model
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ABSTRACT. We consider the ferromagnetic $q$-state Potts model on the
$d$-dimensional lattice $\integ ^d,\;d\geq 2$.
Suppose that the Potts variables $(\rho _x,\; x\in \integ
^d)$ are distributed in one of the $q$ low temperature
phases. Suppose that $n\not=1,q$ divides $q$.
Partitioning the single site state space into $n$ equal
parts $K_1,\ldots ,K_n$, we obtain a new random field
$\sigma =(\sigma _x,\; x\in \integ ^d)$ by defining fuzzy variables
$\sigma _x=\alpha $ if $\rho _x\in K_{\alpha },\;
\alpha =1,\ldots ,n$. We investigate the state induced
on these fuzzy variables. First we look at the conditional
distribution of $\rho _x$ given all values $\sigma _y, y\in \integ ^d$.
We find that below the critical temperature all versions of this
conditional
distribution are non-quasilocal on a set of configurations which carries
positive measure. Then we look at the
conditional distribution of $\sigma _x$ given all values $\sigma _y, y\not=x$.
If the system is not at the critical temperature of a first order phase
transition, there exists a version of this conditional distribution that is
almost surely quasilocal.