1$ for some $i\in \{1, \ldots, s\}$, and there is no loss of generality in assuming that $r_1>1$. If $| {\cal G}|>1$, then $\pi^{G,1}_{q, \beta} \neq \pi^{G,0}_{q, \beta}$ due to (\ref{eq:positive_magnetization}) and (\ref{eq:zero_magnetization}). Hence, using Proposition \ref{prop:wired}, $\mu^{G,1}_{q, \beta, (r_1, \ldots, r_s)}$ is nonquasilocal and therefore non-Gibbsian, and the `if' part of the theorem is established. For the `only if' part, note that if $| {\cal G}|=1$, then ${\cal G}=\{\pi^{G,0}_{q, \beta}\}$, so that $\mu^{G,0}_{q, \beta, (r_1, \ldots, r_s)}$ is the only fuzzy Potts measure, which by Proposition \ref{prop:free} is quasilocal and therefore Gibbsian. $\Cox$ \medskip\noindent It remains to prove Propositions \ref{prop:free} and \ref{prop:wired}. To this end, we need to introduce the notion of a tree-indexed Markov chain on $\Gamma$, and its relation to Gibbs measures for the Potts model on $\Gamma$. This relation is well-known for regular trees (see for instance \cite{Sp,Z,BW}), while the extension to general trees seems to be less well-studied. Let $(x_0, x_1, \ldots)$ be an enumeration of $V_{\Gamma}$ such that the root $\rho$ comes first $(x_0=\rho$), then all vertices in $V_{\Gamma_1} \setminus \{\rho\}$, then all vertices in $V_{\Gamma_2} \setminus V_{\Gamma_1}$, and so on. Fix $q$, let $\nu$ be a probability measure on $\{1, \ldots, q\}$ (which will play the role of an initial distribution), and let $P = (P_{ij})_{i,j \in \{1, \ldots, q\}}$ be a transition matrix. Let $X$ be the $\{1, \ldots, q\}^{V_{\Gamma}}$-valued random spin configuration obtained as follows. First pick $X(x_0)\in \{1, \ldots, q\}$ according to $\nu$. Then, inductively, once $X(x_0), \ldots, X(x_n)$ have been determined, pick $X(x_{n+1})\in \{1,\ldots, q\}$ with distribution $(P_{i1}, \ldots, P_{iq})$ where $i=X(\parent(x_{n+1}))$. For obvious reasons, $X$ is called a tree-indexed Markov chain on $\Gamma$. There is sometimes reason to consider inhomogeneous tree-indexed Markov chains, where the transition matrix $P$ is allowed to depend on where in the tree we are: for every $x\in V_\Gamma \setminus \{ \rho \}$, we then have a transition matrix $P^x = (P^x_{ij})_{i,j \in \{1, \ldots, q\}}$, and $X$ is generated as above with $X(x)$ chosen according to the distribution $(P^x_{i1}, \ldots, P^x_{iq})$ where $i=X(\parent(x))$. It is readily checked that a (possibly inhomogeneous) tree-indexed Markov chain $X$ is also a Markov random field on $\Gamma$, meaning that for any finite $W \subset V_\Gamma$, the conditional distribution of $X(W)$ given $X(V_\Gamma \setminus W)$ depends on $X(V_\Gamma \setminus W)$ only via $X(\partial W)$. Hence the supremum in (\ref{eq:def_quasilocality}) becomes $0$ for all $n$ large enough so that $\Lambda_n$ contains $W \cap \partial W$, so that we have the following lemma. \begin{lem} \label{lem:chains_are_quasilocal} The distribution of any homogeneous or inhomogeneous tree-indexed Mar\-kov chain on $\Gamma$ is quasilocal. \end{lem} Fix $\beta\geq 0$, and consider the tree-indexed Markov chain given by $\nu=(\frac{1}{q}, \ldots, \frac{1}{q})$ and transition matrix $P=(P_{ij})_{i,j \in \{1, \ldots, q\}}$ given by \begin{equation} \label{eq:transition_matrix} P_{ij} = \left\{ \begin{array}{ll} \frac{e^{2\beta}}{e^{2\beta}+q-1} & \mbox{if } i =j \\ \frac{1}{e^{2\beta}+q-1} & \mbox{otherwise.} \end{array} \right. \end{equation} Let $X\in\{1, \ldots, q\}^{V_{\Gamma}}$ be given by this particular tree-indexed Markov chain. By directly checking (\ref{eq:Potts}), we see that $X(\Lambda_n)$ has distribution $\pi^{\Gamma_n}_{q,\beta}$. By taking limits as $n\rightarrow \infty$ and considering the construction of $\pi^{G,0}_{q,\beta}$ in Section \ref{subsect:Potts_infinite}, we see that $X$ is distributed according to the Gibbs measure $\pi^{\Gamma,0}_{q,\beta}$ for the Potts model on $\Gamma$ with free boundary condition. \medskip\noindent {\bf Proof of Proposition \ref{prop:free}:} Construct $X\in \{1, \ldots, q\}^{V_{\Gamma}}$ sequentially as above, with $\nu= (\frac{1}{q}, \ldots, \frac{1}{q})$ and $P$ given by (\ref{eq:transition_matrix}), and let $Y\in \{1, \ldots, r\}^{V_{\Gamma}}$ from $X$ as in (\ref{eq:def_of_fuzzy_Potts}). Then the conditional distribution of $Y(x_{n+1})$ given $X(x_0), \ldots, X(x_n)$ such that $X(\parent(x_{n+1}))=i$ and $Y(\parent(x_{n+1}))=k)$, is given by \begin{equation} \label{eq:Y_is_a_Markov_chain} \P(Y(x_{n+1}) = l \, | \, \cdots) = \left\{ \begin{array}{ll} \frac{e^{2\beta}+r_k-1}{e^{2\beta}+q-1} & \mbox{if } l=k \\ \frac{r_k}{e^{2\beta}+q-1} & \mbox{otherwise,} \end{array} \right. \end{equation} which follows by summing over the possible values of $X(x_{n+1})$. Note that the right-hand side of (\ref{eq:Y_is_a_Markov_chain}) depends on $X(x_0), \ldots, X(x_n)$ only through $Y(\parent(x_{n+1}))$. It follows that $Y$ is a tree-indexed Markov chain with state space $\{1, \ldots, s\}$, initial distribution $(\frac{r_1}{q}, \ldots, \frac{r_s}{q})$ and transition matrix $P=(P_{kl})_{k,l \in \{1, \ldots, s\}}$ given by \begin{equation} \label{Markov_chain_for_Y_with_free_bc} P_{kl} = \left\{ \begin{array}{ll} \frac{e^{2\beta}+r_l-1}{e^{2\beta}+q-1} & \mbox{if } l=k \\ \frac{r_l}{e^{2\beta}+q-1} & \mbox{otherwise.} \end{array} \right. \end{equation} Quasilocality of $Y$ now follows from Lemma \ref{lem:chains_are_quasilocal}. $\Cox$ \medskip\noindent For the proof of Proposition \ref{prop:wired}, we need to consider the tree-indexed Markov chain on $\Gamma$ corresponding to the Gibbs measure $\pi^{\Gamma,1}_{q, \beta}$ with the ``all $1$'' boundary condition. This is a bit more complicated than the case of $\pi^{\Gamma,0}_{q, \beta}$ due to the lack of full symmetry among the spin values. For $x\in V_{\Gamma}$, consider the Gibbs measure $\pi^{\Gamma_{(x)},1}_{q, \beta}$, and in particular the probability $\pi^{\Gamma_{(x)},1}_{q, \beta}(\mbox{spin $1$ at } x)$, which we denote by $a_x$. (Note that $a_x$ is in general distinct from $\pi^{\Gamma,1}_{q, \beta}(\mbox{spin $1$ at } x)$, because it fails to take into account, e.g., the possible influence from $\parent(x)$ on $x$.) For symmetry reasons, the $\pi^{\Gamma_{(x)},1}_{q, \beta}$-distribution of the spin at $x$ is \[ \left(a_x, \frac{1-a_x}{q-1}, \frac{1-a_x}{q-1}, \ldots, \frac{1-a_x}{q-1} \right) \, . \] Also define \begin{equation} \label{eq:def_b_x} b_x = \frac{a_x}{(1-a_x)/(q-1)} = \frac{\pi^{\Gamma_{(x)},1}_{q, \beta}(\mbox{spin $1$ at } x)}{ \pi^{\Gamma_{(x)},1}_{q, \beta}(\mbox{spin $2$ at } x)} \, . \end{equation} The constants $\{b_x\}_{x\in V_\Gamma}$ satisfy the following recursion. \begin{lem} \label{lem:wired_recursion} Suppose $x \in V_\Gamma$ is a vertex with $k$ children $y_1 , \ldots, y_k$. We then have \begin{equation} \label{eq:wired_recursion} b_x = \frac{\prod_{i=1}^k (e^{2\beta} b_{y_i} + q - 1)}{ \prod_{i=1}^k (e^{2\beta} + b_{y_i} + q - 2)} \, . \end{equation} \end{lem} {\bf Proof:} For $n$ large enough so that $x \in V_{\Lambda_n}$, define, as a finite-volume analogue of (\ref{eq:def_b_x}), \[ b_{x,n} = \frac{\pi^{\Gamma_{(x,n)},1}_{q, \beta}(\mbox{spin $1$ at } x)}{ \pi^{\Gamma_{(x,n)},1}_{q, \beta}(\mbox{spin $2$ at } x)} \, , \] where $\pi^{\Gamma_{(x,n)},1}_{q, \beta}$ is the finite-volume Gibbs measure for $\Gamma_{(x,n)}$ with spin $1$ boundary condition on those vertices sitting furthest away from $x$ in $\Gamma_{(x,n)}$, i.e., those at distance $n$ from $\rho$ in $\Gamma$. By the construction of Gibbs measures in Section \ref{subsect:Potts_infinite}, we have \begin{equation} \label{b's_as_limits} \lim_{n\rightarrow \infty} b_{x,n} = b_x \, . \end{equation} Imagine now the modified graph $\Gamma^*_{(x,n)}$ obtained from $\Gamma_{(x,n)}$ by removing all edges incident to $x$. In other words, $\Gamma^*_{(x,n)}$ is a disconnected graph with an isolated vertex $x$ together with $k$ connected components isomorphic to $\Gamma_{(y_1,n)}, \ldots, \Gamma_{(y_k,n)}$. When picking $X \in \{1, \ldots, q\}^{V_{\Gamma^*_{(x,n)}}}$ according to $\pi^{\Gamma^*_{(x,n)},1}_{q, \beta}$, the spin configurations on different connected components obviously become independent. In particular, if we only consider the spins $(X(x), X(y_1), \ldots, X(y_k))$, then we can note that these spins become independent, with $X(x)$ having distribution $(\frac{1}{q}, \ldots, \frac{1}{q})$, and $X(y_i)$ having distribution $(\frac{b_{y_i, n}}{b_{y_i, n}+q-1}, \frac{1}{b_{y_i, n}+q-1}, \ldots, \frac{1}{b_{y_i, n}+q-1})$. If we now reinsert the edges between $x$ and $y_1 , \ldots, y_k$, thus recovering $\Gamma_{(x,n)}$, then the $\pi^{\Gamma_{(x,n)},1}_{q, \beta}$-distribution of $(X(x), X(y_1), \ldots, X(y_k))$ becomes the same as the corresponding $\pi^{\Gamma^*_{(x,n)},1}_{q, \beta}$-distribution above except that each configuration $\xi\in\{1, \ldots, q\}^{\{x, y_1, \ldots, y_k\}}$ is reweighted by a factor $\exp(2\beta\sum_{i=1}^k I_{\{\xi(y_i)=\xi(x)\}})$. Hence \[ \pi^{\Gamma_{(x,n)},1}_{q, \beta}((X(x), X(y_1), \ldots, X(y_k))=\xi) =\frac{1}{Z} \prod_{i=1}^k (e^{2\beta I_{\{\xi(y_i)=\xi(x)\}}} \,b_{y_i,n}^{I_{\{\xi(y_i)=1\}}}) \] for some normalizing constant $Z$. By integrating out $X(y_1), \ldots, X(y_k)$, we get \[ b_{x,n} = \frac{\prod_{i=1}^k (e^{2\beta} b_{y_i,n} + q - 1)}{ \prod_{i=1}^k (e^{2\beta} + b_{y_i,n} + q - 2)} \, . \] Sending $n\rightarrow \infty$ in this expression, and using (\ref{b's_as_limits}) $k+1$ times (substituting $x$ with itself and with $y_1, \ldots, y_k$), we obtain (\ref{eq:wired_recursion}), as desired. $\Cox$ \medskip\noindent Note that the above proof yields that given $X(x)=1$, the spins $X(y_1), \ldots, X(y_k)$ become conditionally independent, with $X(y_i)$ having distribution \[ \left(\frac{b_{y_i}e^{2\beta}}{b_{y_i}e^{2\beta} + q-1}, \frac{1}{b_{y_i}e^{2\beta} + q-1}, \ldots, \frac{1}{b_{y_i}e^{2\beta} + q-1}\right) \, . \] Likewise, for $l \neq 1$, conditioning on $X(x)=l$ makes $X(y_1), \ldots, X(y_k)$ conditionally independent with $X(y_i)$ taking value $1$ with probability $\frac{b_{y_i}}{b_{y_i}+ e^{2\beta}+q-2}$, value $l$ with probability $\frac{e^{2\beta}}{b_{y_i}+ e^{2\beta}+q-2}$, and other values with probabilities $\frac{1}{b_{y_i}+ e^{2\beta}+q-2}$. By iterating the above argument, we arrive at the following tree-indexed Markov chain description of the Gibbs measure $\pi^{\Gamma,1}_{q, \beta}$. \begin{lem} \label{lem:wired_as_a_Markov_chain} Suppose that the random spin configuration $X\in \{1, \ldots, q\}^{V_\Gamma}$ is obtained as an inhomogeneous tree-indexed Markov chain with initial distribution \[ \nu=\left(\frac{b_\rho}{b_\rho + q-1}, \frac{1}{b_\rho + q-1} \ldots, \frac{1}{b_\rho + q-1} \right) \] and transition matrices $P^x = (P^x_{ij})_{i,j \in \{1, \ldots, q\}}$ given by \[ P^x_{ij} = \left\{ \begin{array}{ll} \frac{b_x e^{2\beta}}{b_x e^{2\beta} + q-1} & \mbox{if } i=j=1 \\ \frac{1}{b_x e^{2\beta} + q-1} & \mbox{if } i=1, \, j \neq 1 \\ \frac{b_x}{b_x+ e^{2\beta}+q-2} & \mbox{if } i\neq 1, j =1 \\ \frac{e^{2\beta}}{b_x+ e^{2\beta}+q-2} & \mbox{if } i=j \neq 1 \\ \frac{1}{b_x+ e^{2\beta}+q-2} & \mbox{otherwise.} \end{array} \right. \] Then $X$ has distribution $\pi^{\Gamma,1}_{q, \beta}$. \end{lem} A crucial difference now compared to the Gibbs measure $\pi^{\Gamma,0}_{q, \beta}$ with free boundary condition is that if any $b_x \neq 1$, then there is not enough state-symmetry in the tree-indexed Markov chain in Lemma \ref{lem:wired_as_a_Markov_chain} to make the corresponding fuzzy Potts model a tree-indexed Markov chain. This will soon become clear. A key lemma for proving nonquasilocality in the fuzzy Potts model is the following. \begin{lem} \label{lem:sibling} If $\pi^{\Gamma,1}_{q, \beta} \neq \pi^{\Gamma,0}_{q, \beta}$, then there exist two siblings $y_1, y_2 \in V_\Gamma$ such that $b_{y_i}>1$ for both $i=1$ and $i=2$. \end{lem} {\bf Proof:} It follows from the assumption $\pi^{\Gamma,1}_{q, \beta} \neq \pi^{\Gamma,0}_{q, \beta}$ using (\ref{eq:positive_magnetization}) that $a_\rho>\frac{1}{q}$, so that \begin{equation} \label{eq:magnetization_at_rho} b_\rho>1 \, . \end{equation} Furthermore, (\ref{eq:positive_magnetization}) and (\ref{eq:zero_magnetization}) imply that $a_x \geq \frac{1}{q}$ for all $x\in V_\Gamma$, whence $b_x \geq 1$ for all $x\in V_\Gamma$. Note also that $1$ is a fixed point of the recursion (\ref{eq:wired_recursion}), in the sense that if all children $y_1, \ldots, y_k$ satisfy $b_{y_i}=1$, then $b_x=1$. Hence, $\rho$ must have at least one child $x$ with $b_x>1$. By iterating this argument we see that for any $n$, it must have at least one descendant $x$ at distance $n$ such that $b_x>1$. Fix $n$ and such a vertex $x$ with $b_x>1$ at distance $n$ from $\rho$. Write $(z_0, z_1, \ldots, z_n)$ for the vertices on the self-avoiding path from $x$ to $\rho$ (so that in particular $z_0=x$ and $z_n = \rho$). Next, note that the recursion (\ref{eq:wired_recursion}) has the property that if one of the children $y_i$ has $b_{y_1}>1$, then $b_x>0$ as well. Since $b_{z_0}>1$ it follows that $b_{z_i}>1$ for $i=1, \ldots, n$. Suppose now for contradiction that the assertion of the lemma is false, i.e., that there are no two siblings $y_1, y_2 \in V_\Gamma$ for which $b_{y_1}>1$ and $b_{y_2}>1$. Then none of the vertices $z_0, \ldots, z_{n-1}$ has a sibling $y$ with $b_y>1$. The recursion (\ref{eq:wired_recursion}) along the path $(z_0, z_1, \ldots, z_n)$ then turns into a simple one-dimensional dynamical system on the space $[1, \infty)$ given by $b_{z_{i+1}}= f(b_{z_i})$ where \[ f(b) = \frac{e^{2\beta}b+q-1}{e^{2\beta}+b+q-2} \, . \] This dynamical system is contractive with a unique fixed point at $b=1$, so that -- if we just keep iterating beyond the $n$'th iteration -- for any initial value $b_{z_0}\in [1, \infty)$ we obtain \begin{equation} \label{eq:limit_of_dynamical_system} \lim_{n \rightarrow \infty} b_{z_n}=1 \, . \end{equation} Since $f$ is increasing and bounded by $e^{2\beta}$, we get that $b_{z_1}$ is bounded by $e^{2\beta}$ and, therefore, that the convergence in (\ref{eq:limit_of_dynamical_system}) is in fact uniform in the initial value $b_{z_0}$. Thus we can, for any $\eps>0$, find an $n$ which guarantees that $b_{z_n}<1+ \eps$. Thus, $b_\rho <1+ \eps$ for any $\eps>0$, whence $b_\rho=1$. But this contradicts (\ref{eq:magnetization_at_rho}), so the proof is complete. $\Cox$ \medskip\noindent {\bf Proof of Proposition \ref{prop:wired}:} By Lemma \ref{lem:sibling}, $\Gamma$ has at least one vertex which has (at least) two children $y_1$ and $y_2$ that both have $b_{y_i}>1$. The choice of root $\rho$ for the tree does not influence the Gibbs measure $\pi^{\Gamma,1}_{q, \beta}$, and therefore we may assume that $\rho$ has two such children $y_1$ and $y_2$. We shall for simplicity first prove the proposition under the assumption that \begin{equation} \label{eq:preliminary_assumption} \mbox{$\rho$ has no other children,} \end{equation} and in the end show how to remove this assumption. We shall have a look at the conditional distribution of the fuzzy spin $Y(\rho)$ at the root, given that its neighbors (i.e., its children) take value \begin{equation} \label{eq:all_1's} Y(y_1)= Y(y_2)=1 \, . \end{equation} By summing over all $X\in \{1, \ldots, q\}^{\{\rho, y_1, y_2\}}$ such that (\ref{eq:all_1's}) holds, and using Lemma \ref{lem:wired_as_a_Markov_chain}, we obtain \begin{eqnarray} \label{eq:crucial_ratio_of_probabilities} \lefteqn{\frac{\P(Y(\rho)=1 \, | \, Y(y_1) = Y(y_2)=1)}{ \P(Y(\rho) \neq 1 \, | \, Y(y_1) = Y(y_2)=1)}} \\ \nonumber & = & \frac{\frac{b_\rho}{b_\rho+q-1}\prod_{i=1}^2 \frac{b_{y_i}e^{2\beta}+r_1-1}{{b_{y_i}e^{2\beta}+q-1}} + \frac{r_1-1}{b_\rho+q-1}\prod_{i=1}^2 \frac{b_{y_i}+e^{2\beta}+r_1-2}{{b_{y_i}+e^{2\beta}+q-2}}} {\frac{q-r_1}{b_\rho+q-1} \prod_{i=1}^2 \frac{b_{y_i}+r_1-1}{{b_{y_i}+e^{2\beta}+q-2}}} \\ \nonumber & = & \frac{\prod_{i=1}^2 (b_{y_i}e^{2\beta} + r_1-1)+ (r_1-1)\prod_{i=1}^2 (b_{y_i}+e^{2\beta} + r_1-2)}{(q-r_1) \prod_{i=1}^2 (b_{y_i} +r_1-1)} \end{eqnarray} where in the last line we have used (\ref{eq:wired_recursion}) to express $b_\rho$ in terms of the $b_{y_i}$'s. Now pick an $n$, and consider conditioning further on some $\eta_n \in \{1, \ldots, s\}^{V_{\Gamma_{n+1}} \setminus \{\rho\}}$ such that $\eta_n(y_1) = \eta_n(y_2)=1$. The conditional probability $\P(Y(\rho)=1 \, | \, Y(y_1) = Y(y_2)=1)$ is a convex combination of terms $\P(Y(\rho)=1 \, | \, Y(V_{\Gamma_{n+1}} \setminus \{\rho\}) = \eta_n)$ for such $\eta_n$'s. We can therefore find a particular $\eta_n\in\{1, \ldots, s\}^{\Lambda_{n+1} \setminus \{\rho\}}$ such that \begin{eqnarray} \nonumber \lefteqn{ \frac{\P(Y(\rho)=1 \, | \, Y(V_{\Gamma_{n+1}} \setminus \{\rho\}) = \eta_n)}{ \P(Y(\rho) \neq 1 \, | \, Y(V_{\Gamma_{n+1}} \setminus \{\rho\}) = \eta_n)}} \\ & \geq & \frac{\prod_{i=1}^2 (b_{y_i}e^{2\beta} + r_1-1)+ (r_1-1)\prod_{i=1}^2 (b_{y_i}+e^{2\beta} + r_1-2)}{(q-r_1) \prod_{i=1}^2 (b_{y_i} +r_1-1)} \label{eq:wired_conditioning_on_eta} \, . \end{eqnarray} Fix such an $\eta_n$. Next, construct another configuration $\eta'_n\in \{1, \ldots, s\}^{V_{\Gamma_{n+1}} \setminus \{\rho\}}$ by taking \[ \eta'_n(x) = \left\{ \begin{array}{ll} \eta_n(x) & \mbox{for } x \in V_{\Gamma_n} \setminus \{\rho\} \\ (\eta_n(\parent(x))+1) \, {\rm mod} \, s & \mbox{for } x \in V_{\Gamma_{n+1}} \setminus V_{\Gamma_n} \, . \end{array} \right. \] The crucial aspects of this choice of $\eta'_n$ is that (a) $\eta_n=\eta'_n$ on $V_{\Gamma_n}$ and (b) each $x$ in the remotest layer $V_{\Gamma_{n+1}}\setminus V_{\Gamma_n}$ of $\Gamma_{n+1}$ has a fuzzy spin value which is different from its parent. It is readily checked that property (b) implies that the conditional distribution of $Y(V_{\Gamma_{n-1}})$ given $Y(V_{\Gamma_{n+1}}\setminus V_{\Gamma_{n-1}})= \eta'_n(V_{\Gamma_{n+1}}\setminus V_{\Gamma_{n-1}})$ becomes the same as if the underlying Gibbs measure had been not $\pi^{\Gamma,1}_{q, \beta}$ but rather the finite-volume Gibbs measure $\pi^{\Gamma_{n+1}}_{q, \beta}$ (cf.\ \cite[Lem.\ 9.2]{H3}). Hence the conditional distribution of $Y(\rho)$ given that $Y(V_{\Gamma_{n+1}} \setminus \{\rho\}) = \eta'_n)$ can be calculated from the tree-indexed Markov chain corresponding to free boundary condition, i.e., the one defined in (\ref{Markov_chain_for_Y_with_free_bc}). We get \begin{equation} \label{eq:free_conditioning_on_eta} \frac{\P(Y(\rho)=1 \, | \, Y(V_{\Gamma_{n+1}} \setminus \{\rho\}) = \eta'_n)}{ \P(Y(\rho) \neq 1 \, | \, Y(V_{\Gamma_{n+1}} \setminus \{\rho\}) = \eta'_n)} = \frac{(e^{2\beta}+r_1-1)^2}{(q-r_1)r_1} \end{equation} Note that the right-hand sides of (\ref{eq:wired_conditioning_on_eta}) and (\ref{eq:free_conditioning_on_eta}) do not depend on $n$. We now make the following crucial claim. \begin{quote} {\bf Claim:} If $b_{y_1}>1$ and $b_{y_2}>1$, then the right-hand side of (\ref{eq:wired_conditioning_on_eta}) is strictly greater than the right-hand side of (\ref{eq:free_conditioning_on_eta}). \end{quote} To prove the claim, define \[ a = \frac{b_{y_1}b_{y_2}+r_1 - 1}{(b_{y_1}+r_1 - 1)(b_{y_2} + r_1 - 1)} \] and note that $a$ can be rewritten as \begin{eqnarray*} a & = & \frac{b_{y_1}b_{y_2}+r_1 - 1}{(b_{y_1}+r_1 - 1)(b_{y_2} + r_1 - 1)} \\ & = & \frac{1}{r_1}\frac{r_1}{(b_{y_1}+r_1-1)} + \frac{b_{y_2}}{(b_{y_2}+r_1 - 1)} \frac{(b_{y_1}-1)}{(b_{y_1}+r_1-1)} \, . \end{eqnarray*} Assuming that $b_{y_1}>1$ and $b_{y_2}>1$, we get that $\frac{b_{y_2}}{b_{y_2}+r_1 - 1} > \frac{1}{r_1}$ and that $\frac{b_{y_1}-1}{b_{y_1}+r_1-1}>0$, whence \begin{eqnarray} \nonumber a & > & \frac{1}{r_1}\frac{r_1}{(b_{y_1}+r_1-1)} + \frac{1}{r_1} \frac{(b_{y_1}-1)}{(b_{y_1}+r_1-1)} \\ & = & \frac{1}{r_1} \, . \label{eq:inequality_for_a} \end{eqnarray} Next, an elementary but tedious calculation shows that the right-hand side of (\ref{eq:wired_conditioning_on_eta}) can be rewritten as \begin{equation} \label{eq:rewritten_wired} \frac{a(e^{4\beta}+r_1-1) + (1-a) (2e^{2\beta} + r_1 - 2)}{q - r_1} \, . \end{equation} Analogously, the right-hand side of (\ref{eq:free_conditioning_on_eta}) can be rewritten as \begin{equation} \label{eq:rewritten_free} \frac{\frac{1}{r_1}(e^{4\beta}+r_1-1) + (1-\frac{1}{r_1}) (2e^{2\beta} + r_1 - 2)}{q - r_1} \, . \end{equation} Now, using (\ref{eq:inequality_for_a}) and the observation that \[ e^{4\beta}+r_1-1 > 2e^{2\beta} + r_1 - 2 \, , \] we get that the expression in (\ref{eq:rewritten_wired}) is strictly greater than that in (\ref{eq:rewritten_free}), and the claim is proved. Hence the difference between the left-hand sides of (\ref{eq:wired_conditioning_on_eta}) and (\ref{eq:free_conditioning_on_eta}) is bounded away from $0$ uniformly in $n$. The denominators of the left-hand sides are bounded away from $0$ uniformly in $n$ due to uniform nonnullness of the fuzzy Potts model (see Section \ref{sect:quasilocal}). Hence \[ \P(Y(\rho)=1 \, | \, Y(V_{\Gamma_{n+1}} \setminus \{\rho\}) = \eta_n) - \P(Y(\rho)=1 \, | \, Y(V_{\Gamma_{n+1}} \setminus \{\rho\}) = \eta'_n) \] is bounded away from $0$ uniformly on $n$. By plugging in these $\eta_n$ and $\eta'_n$ in (\ref{eq:def_quasilocality}), we get, since $\eta_n=\eta'_n$ on $V_{\Gamma_n}$, that quasilocality of $Y$ fails. This proves the proposition modulo the assumption (\ref{eq:preliminary_assumption}). It remains to remove the assumption (\ref{eq:preliminary_assumption}). To do this, suppose that $\rho$ has $k-2$ additional children $y_3, \ldots, y_k$. We can then extend the configurations $\eta$ and $\eta'$ that we condition on above, to $y_3, \ldots, y_k$ and their descendants, as follows. We insist that $\eta$ and $\eta'$ that they take value $1$ at $y_3, \ldots, y_k$, and that they take some value other than $1$ at all children of $y_3, \ldots, y_k$ (they may otherwise be arbitrary on the further descendants of $y_3, \ldots, y_k$). Easy modifications of the calculations above show that (\ref{eq:wired_conditioning_on_eta}) and (\ref{eq:free_conditioning_on_eta}) hold as before, with the modification that both right-hand sides are multiplied by \[ \left( \frac{e^{2\beta} + r_1 - 1}{r_1} \right)^{k-2} \, . \] Since this factor is the same in (\ref{eq:wired_conditioning_on_eta}) and (\ref{eq:free_conditioning_on_eta}), the rest of the proof goes through as before. $\Cox$ \medskip\noindent {\bf Remark:} Since the event conditioned on in (\ref{eq:crucial_ratio_of_probabilities}) has positive measure, it is easy to extract from the above proof that the set of discontinuities of the conditional probability $\P(Y(\rho)=1 | Y(V_\Gamma\setminus \{\rho\}) = \eta)$ as a function of $\rho$, has positive measure under $\mu^{G,1}_{q, \beta, (r_1, \ldots, r_s)}$. Hence, so-called almost sure quasilocality and almost sure Gibbsianness fails in general for the fuzzy Potts model on trees, in contrast to the $\Z^d$ case (see Maes and Vande Velde \cite{MVV}) and the mean-field case (Theorem \ref{thm:main_result_on_complete_graph} (iv)). This contrast between the fuzzy Potts model on $\Z^d$ and on trees is analogous to the corresponding almost sure Gibbsianness issue for the random-cluster model; see \cite{H96}. \subsection{Discussion} What concrete information can we extract from Theorem \ref{thm:main_result_on_trees}? Let $\beta_c=\beta_c(\Gamma,q)$ denote, as in Section \ref{subsect:Potts_infinite}, the critical value for the $q$-state Potts model on the tree $\Gamma$. For $q\geq 3$, we then have from Theorem \ref{thm:main_result_on_trees} that $\beta<\beta_c$ implies that any corresponding fuzzy Potts measure is Gibbsian, while $\beta>\beta_c$ yields existence of corresponding fuzzy Potts measures that are non-Gibbsian. It remains to specify the critical value $\beta_c(\Gamma,q)$. If we know the critical value $p_c(\Gamma,q)$ of the corresponding random-cluster model, then we can calculate $\beta_c= -\frac{1}{2} \log (1-p_c)$ (see, e.g., \cite{GHM}). For the case when $\Gamma$ is a regular tree, the critical value $p_c(\Gamma,q)$ can be characterized in terms of the solutions of a certain algebraic equation given in \cite[p.\ 235]{H96a}. For general trees the situation is more complicated. For a variety of stochastic models on trees, critical values can be calculated in terms of a natural quantity known as the branching number of the tree, denoted ${\rm br}(\Gamma)$; see for instance \cite{P}. Lyons \cite{L} calculated $\beta_c(\Gamma,q)$ in terms of ${\rm br}(\Gamma)$ for the case $q=2$. In contrast, and perhaps somewhat surprisingly, the critical values $\beta_c(\Gamma,q)$ for larger $q$ do {\em not} admit a characterization in terms of ${\rm br}(\Gamma)$; this was shown by Pemantle and Steif \cite{PS}. Bounds for $\beta_c(\Gamma,q)$ that only depend on ${\rm br}(\Gamma)$ and on $q$ can, however, be obtained using the standard comparison techniques for the random-cluster model reviewed in \cite{GHM}. \section{The fuzzy Potts model on complete graphs} \label{sect:mean-field} In this section we treat the case of complete graphs. We start with precise definitions of the model and a detailed explanation of the limiting process for the conditional probabilities that was sketched in the introduction. The proofs are essentially self-contained but use some standard knowledge (whose main reference is Ellis and Wang \cite{ElWa90}) on the infinite volume limit of the empirical distribution of the order parameter in the mean-field Potts model. \subsection{Mean-field Potts in finite volume $N$} For a positive integer $q$, the Gibbs measure $\pi^N_{q,\beta}$ for the $q$-state Potts model on the complete graph with $N$ vertices at inverse temperature $\beta\geq 0$, is the probability measure on $\{1, \ldots, q\}^N$ which to each $\xi\in \{1, \ldots, q\}^N$ assigns probability \begin{equation} \label{eq:Potts-mf} \pi^N_{q,\beta}(\xi)=\frac{1}{Z^N_{q, \beta}}\exp\left( \frac{\beta}{N} \sum_{1\leq x\neq y \leq N} I_{\{\xi(x) = \xi(y)\}} \right) \, . \end{equation} Here $Z^N_{q, \beta}$ is the normalizing constant. Note that this definition slightly deviates from the definition (\ref{eq:Potts}) by the factor $1/N$ appearing in the exponential. Such a convention is appropriate because, clearly, the interaction must be chosen depending on the size of the graph in a mean-field model. This definition of the finite volume Gibbs-measures is standard in the literature; see e.g.\ \cite{ElWa90}. %When %ÔÔNon-GibbsiannessÕÕ for mean-field models is understood as ÔÔdiscontinuity %of conditional probabilities as a function of the conditioning,ÕÕ it becomes a %meaningful and natural notion. Of course the notion of continuity has to %be taken in the appropriate sense. Indeed, the Gibbs measures of simple %mean-field models (like the standard Curie Weiss model) usually converge %weakly to linear combinations of product measures. A (non-trivial) linear %combination of product measures is non-Gibbsian and has each spin configuration %as a discontinuity point.(10) So one could feel discouraged to look %for non-trivial continuity properties in conditional probabilities of mean-field %models. In contrast to that, the proceeding that is appropriate for our %mean-field model is as follows: (1) Take the conditioning while staying in %finite volume. (2) Observe that the conditional probabilities outside a finite %set are automatically volume-dependent functions of the empirical average %over all the (joint) spins in the conditioning. (3) Derive the large volume-asymptotics %for these functions. (4) Consider their continuity properties. %Look at the size of the set of their discontinuity points in the large volume-limit. \subsection{Mean-field fuzzy Potts in finite volume $N$} The mean-field fuzzy Potts measure in finite volume $N$ is then defined in the same way as it is defined on every graph. To be explicit, fix $q$, $\beta$ and the spin-partition $(r_1, \ldots, r_s)$ as above. Let $X$ be the $\{1, \ldots, q\}^N$-valued random object distributed according to the mean field finite volume Gibbs measure $\pi^N_{q, \beta}$. Take $Y$ to be the $\{1\ldots, s\}^N$-valued random object obtained from $X$ by the site-wise application of the spin-partitioning as in (\ref{eq:def_of_fuzzy_Potts}). Then $\mu^N_{q, \beta, (r_1, \ldots, r_s)}$ is the probability measure on $\{1, \ldots, s\}^N$ which describes the distribution of $Y$. \subsection{Gibbsianness vs.\ non-Gibbsianness for mean-field models: \\ continuity vs.\ discontinuity of limiting conditional probabilities} We start with some general remarks about mean-field models to explain the appropriate analogue of non-Gibbsianness in more detail than we did in the introduction. To begin with, the following lemma makes explicit that we can always describe the single-site conditional probabilities of the finite volume Gibbs measures of a mean-field model in terms of a {\it single-site kernel} from the empirical distribution vector of the conditioning to the single-site state space. It is the infinite volume limit of this kernel that shall then be considered in the analysis of the model. So, suppose that $S$ is a finite set (local spin space) and for any $N$ we are given an exchangeable (that is permutation-invariant) measure $\mu^N$ on $S^N$. This permutation invariance is certainly true for the mean-field Potts model. Moreover it carries over trivially to the fuzzy Potts model. This is clear since the distribution of the latter is simply obtained by an application of the same map to the spin variable at each site. In a general context, denote by $\PP=\{ (p_i)_{i\in S}, 0\leq p_i\leq 1, \sum_{i\in S}p_i =1 \}$ the space of probability vectors on the set $S$. We use the obvious short notation $x^c=\{1,\dots, N\}\backslash \{x\}$. \begin{lem} \label{lem:kernel} For each $N$ there is a probability kernel $Q^N:S\times\PP\rightarrow [0,1]$ from $\PP$ to the single-site state space $S$ such that the single-site conditional expectations at any site $x$ can be written in the form \begin{equation} \label{eq:0.1} \mu^N\bigl(X(x)=i\bigl | X(x^c)=\eta \bigr)=Q^N \bigl(i\bigl | (n_j)_{j\in S} \bigr) \, . \end{equation} Here $n_j=\frac{1}{N-1}\#\bigl(1\leq y\leq N, y\neq x, \eta(x)=j\bigr)$ is the fraction of sites for which the spin-values of the conditioning are in the state $j\in S$. \end{lem} \noindent {\bf Proof:} By exchangeability it is clear that the right hand side of (\ref{eq:0.1}) depends on the sets $\bigl\{1\leq y\leq N, y\neq x, \eta(y)=j\bigr\}$, for all $j\in S$, only through their size. Equivalently we may express this dependence in terms of the empirical distribution $(n_j)_{j\in S}$. $\Cox$ \medskip\noindent In turn, the knowledge of the kernel $Q^N$ uniquely determines the measure $\mu^N$. This is clear since the knowledge of all one-site conditional probabilities of finitely many random variables uniquely determines the joint distribution. So we may as well consider the $Q^N$«s as the basic objects and regard them as the starting point of the definition of a mean field model. This is of course only meaningful if the $Q^N$«s are related to each other in a meaningful way. Let us turn now to the concrete case of the mean-field Potts model to point out two very simple observations that shall serve as a motivation of our further investigation. In this case we have directly from the definition (\ref{eq:Potts-mf}) the explicit formula \begin{equation} \label{eq:0.2} Q^N_{q,\b} \bigl(i\bigl | (n_j)_{1\leq j\leq q} \bigr)=\frac{\exp\bigl( \b (1-\frac{1}{N})n_i \bigr)}{ \sum_{j=1}^q \exp\bigl( \b (1-\frac{1}{N})n_j \bigr) } \, . \end{equation} We note the following. \begin{description} \item{\bf (i)} $Q^N_{q,\b}$ converges to $Q^\infty_{q,\b} =\frac{\exp\bigl( \b n_i \bigr)}{ \sum_{j=1}^q \exp\bigl( \b n_j \bigr)}$ when $N$ tends to infinity. Indeed, the trivial $1/N$-factor appearing in (\ref{eq:0.2}) could of course even be removed by a harmless redefinition of the model that would lead to the same infinite volume behavior of the Gibbs measures, making all $Q^N_{q,\b}$ identical. \item {\bf (ii)} The limiting kernel $Q^\infty_{q,\b}$ is a {\it continuous function} of the probability vector $(n_j)_{1\leq j\leq q}$, as a function on $\R^q$. \end{description} The existence of the infinite volume limit (i) is a minimal ingredient for the definition of a mean-field model. Assuming this we can talk about limiting or ``infinite volume'' conditional probabities. Then, {\it continuous dependence of the limiting conditional probability} as it is stated in (ii) is the obvious analogue to the continuous dependence of the conditional expectation of a lattice model on the conditioning with respect to product topology. So, properties (i) and (ii) are the analogues of a proper Gibbsian structure for mean-field models. ``Non-Gibbsianness'' may then manifest itself by the failure of (ii) at certain points of discontinuity. The reader may find a number of examples of this in \cite{K3}. After these introductory remarks we will show in the following that discontinuities in fact occur for the mean-field fuzzy Potts model, for certain values of the parameters, and discuss them in detail. \subsection{Conditional probabilities for fuzzy Potts in finite volume} Let us use the following notation for the single-site probability kernel that describes the conditional probabilities of the fuzzy model. \begin{equation} \label{eq:0.3} \mu^N_{q, \beta, (r_1, \ldots, r_s)}\Bigl(Y(x)=k\bigl | Y(x^c)=\eta \Bigr)=:Q^N_{q, \beta, (r_1, \ldots, r_s)} \bigl(k\bigl | (n_l)_{1\leq l\leq s} \bigr) \, . \end{equation} where $n_l=\frac{1}{N-1}\#\bigl(1\leq y\leq N, y\neq x, \eta(x)=l\bigr)$, for $l=1,\dots,s$ is the empirical distribution of fuzzy spin-values in the conditioning. Now, it is not difficult to derive an explicit expression in terms of expectations with respect to ordinary mean-field Potts measures, having the number of states given by the sizes of the classes $r_l$. Clearly, the infinite volume analysis relies on this result. \begin{prop} \label{prop:1}For each finite $N$ we have the representation \begin{equation} \label{eq:prop_1} Q^N_{q, \beta, (r_1, \ldots, r_s)} \bigl(k\bigl | (n_l)_{1\leq l\leq s} \bigr) =\frac{r_{k} \,A(\b_{k},r_{k},N_{k}) }{\sum_{l=1}^{s}r_{l} \,A(\b_{l},r_{l},N_{l}) } \end{equation} where \[ A(\tilde\b,r,M) \equiv \pi^M_{r,\tilde\beta}\Bigl( \exp\Bigl( \frac{\tilde\b}{M} \sum_{x=1}^{M} I_{X(x)=1} \Bigr) \Bigr) \, , \] \[ N_{k} = (N-1)n_k \, , \] and \[ \b_{k} = \frac{\b N_{k}}{N} =\b \Bigl(1-\frac{1}{N}\Bigr)n_k \, . \] \end{prop} \noindent{\bf Remark:} In particular we have $A(\tilde\b,r=1,N)=e^{\tilde\b}$. From this we see immediately that the case of the original Potts model is recovered by setting all $r_l$ equal to one. \medskip\noindent {\bf Proof of Proposition \ref{prop:1}:} To compute the left hand side of (\ref{eq:prop_1}) we may choose $x=1$ and write \begin{eqnarray*} \lefteqn{\textstyle \mu^N_{q, \beta, (r_1, \ldots, r_s)}\Bigl( Y(1)=k\Bigl |Y([2,N])=\eta([2,N])\Bigl)} \\ &=& {\textstyle \frac{1}{\Norm(\eta([2,N])) } \sum_{\xi(1)\mapsto k} \sum_{\xi([2,N])\mapsto \eta([2,N])} \pi^N_{q,\beta}\Bigl(\xi(1),\xi([2,N])\Bigr) \, .} \end{eqnarray*} Here we are summing over Potts configurations $\xi$ that are mapped to the fuzzy Potts configuration $(k,\eta)$ by means of the definition of the fuzzy model given in (\ref{eq:def_of_fuzzy_Potts}). The normalization has to be chosen such that summing over $k=1,\dots,s$ yields one, for each fixed $\eta([2,N])$. The partition function appearing in the Gibbs-average on the right hand side only gives a constant that can be absorbed in the normalization, and so we need only consider \begin{eqnarray*} \lefteqn{\textstyle \sum_{\xi(1)\mapsto k} \sum_{\xi([2,N])\mapsto \eta([2,N])} \exp\Bigl( \frac{\beta}{N} \sum_{1\leq x\neq y \leq N} I_{\{\xi(x) = \xi(y)\}} \Bigr)}\hspace{15mm} \\ &= & {\textstyle \sum_{\xi(1)\mapsto k} \sum_{\xi([2,N])\mapsto \eta([2,N])}\exp\Bigl( \frac{\b}{N}\sum_{2\leq y\leq N} I_{\{\xi(1)=\xi(y)\}} \Bigr)} \\ & & {\textstyle \times \exp\Bigl( \frac{\b}{N}\sum_{2\leq x\neq y \leq N} I_{\{\xi(x) = \xi(y)\}} \Bigr) \, . } \end{eqnarray*} For fixed $\eta([2,N])$ we denote $\L_{l}:= \#\bigl\{ x\in \{2,\dots,N\}:\eta(x)=l \bigr\}$. Then the sum in the last exponential decomposes over these sets, and we can rewrite the right hand side of the last equation in the form \begin{eqnarray*} & \sum_{\xi(1)\mapsto k} \sum_{\xi([2,N])\mapsto \eta([2,N])} \exp\Bigl(\frac{\b_{k}}{N_{k}} \sum_{z\in \L_{k}} I_{\{\xi(z)=\xi(1)\}}\Bigr)\cr &\quad\times \prod_{l=1}^s \exp\Bigl(\frac{\b_{l}} {N_{l}} \sum_{x\b_c(q)\\ \l_0(q)\d_{\frac{1}{q}(1,1,\dots,1)} +\frac{1-\l_0(q)}{q} \sum_{\nu=1}^q\d_{u(\b_c(q),q) \, e_{\nu}+\frac{1-u(\b_c(q),q)}{q}(1,1,\dots,1)} & \mbox{if } \b=\b_c(q) \, , \end{array} \right. \nonumber \end{eqnarray} where $e_{i}$ is the unit vector in the $i$'th coordinate direction of $\R^q$. The quantity $u(\b,q)$ is well defined for $\b\geq \b_c(q)$. It is the largest solution of the mean field equation \begin{eqnarray} \label{eq:meanfield-eq}u=\frac{1-e^{-\b u}}{ 1+(q-1)e^{-\b u} } \end{eqnarray} and obeys the following properties: It is strictly increasing in $\b$, and we have $u(q,\b_c(q))=\frac{q-2}{q-1}$. The constant appearing at the critical point obeys the strict inequality $0<\l_0(q)<1$. \end{thm} Some comments are in order: Obviously, $u(\b,q)$ plays the role of an order parameter. Now, for $\b>\b_c(q)$ the system is in a symmetric linear combination of $\nu$-like states. The limiting empirical distribution becomes the equidistribution on the possible spin values for $\b<\b_c(q)$. It jumps at the critical point for $q\geq 3$. At the critical point itself there is a non-trivial linear combination between both types of measures. To feel comfortable with the mean-field equation (\ref{eq:meanfield-eq}) the reader may note that it is obtained from the equations $n_i=\frac{\exp\bigl( \b n_i \bigr)}{ \sum_{j=1}^q \exp\bigl( \b n_j \bigr)}$ for $i=1,\dots,q$ with the following ansatz: Denote by $i$ the index with the largest $n_j$. Assume that $n_j$ is independent of $j$, for $j\neq i$, and put $u=n_i-n_j$ for some $j\neq i$. Let us mention that the results of Theorem \ref{thm:Empirical measures} can be obtained by a Gaussian transformation and saddle point estimates on the resulting integrals (all of which is omitted here). At the critical point a little care is needed: To obtain the proper value of the constant $\l_0(q)$ a Gaussian approximation around the minima and estimates showing positive curvature are needed. The well-known case of the mean field Ising model $q=2$ can be recovered from the theorem by taking the formal limit $q\downarrow 2$ in the explicit formula for $\b_c(q)$ and noting that $u(q,\b_c(q))=0$. So (\ref{eq:emp-meas}) describes a second order transition in that case. %see Mathematica definition %S[b_, q_] : % FindRoot[x == (1 - Exp[- b x ])/(1 + ( q - 1) Exp[- b x]), {x, 1}][[1]][[2]] %We note that $\b_c(q)$ is a concave function and %strictly increasing for $q\geq 2$. This is seen by the explicit formula. The following explicit formula for the limiting conditional probabilities of the fuzzy model now follows easily from our finite volume representation of the conditional probabilities given in Proposition \ref{prop:1} and the known limiting statement of Theorem \ref{thm:Empirical measures}. \begin{thm}\label{thm:5.4} We have \[ \lim_{N\uparrow \infty}Q^N_{q, \beta, (r_1, \ldots, r_s)} \bigl(k\bigl | (n_l)_{1\leq l\leq s} \bigr) = \frac{C(\b n_{k},r_{k}) }{\sum_{l=1}^{s} C(\b n_{l},r_{l}) } \] whenever $n_{k}\neq \b_{c}(r_{k})/\b$ for all $k$ with $r_{k}\geq 3$. Here \[ C(\tilde\b,r)=\exp\Bigl( \frac{\tilde\b}{r} \Bigr)\times \left\{ \begin{array}{ll} r,& \hbox{if}\,\, \tilde\b<\b_c(r) \cr \exp\Bigl( \frac{\tilde\b(r-1)u(\tilde\b,r)}{r}\Bigr) + (r-1)\exp\Bigl( -\frac{\tilde\b \,u(\tilde\b,r)}{r}\Bigr) ,& \hbox{if} \,\, \tilde\b>\b_c(r) \, . \end{array}\right. \] %&=\exp\Bigl( \frac{\tilde\b}{r} \Bigr)r \times \left\{ %\begin{array}{ll} % 1,& \hbox{if}\,\, \tilde\b<\b_c(r) %\cr % \Bigl(1-u(\tilde\b,r)\Bigr)^{-\frac{r-1}{r}}\Bigl(1+(r-1)u(\tilde\b,r)\Bigr)^{-\frac{1}{r}} % ,& \hbox{if} \,\, \tilde\b>\b_c(r) %\end{array}\right.\cr \end{thm} %The equality of the two expressions follows from the mean-field equation (\ref{eq:meanfield-eq}). \noindent{\bf Proof of Theorem \ref{thm:5.4}:} Let $\tilde \b\neq\b_c(q)$. By Theorem \ref{thm:Empirical measures} we have $\lim_{M\uparrow\infty}r A(\tilde \b,r,M)=C(\tilde \b,r)$. $\Cox$ \medskip\noindent \noindent{\bf Remark:} Obviously this gives the right answer for $\b=0$ or in the case of the original Potts model (letting all $r_l$ be equal to one). We see however that the limiting form of the conditional expectations has a nontrivial form in general. This expression has jumps for $n_{l}=\b_{c}(r_{l})/\b$ whenever $r_{l}\geq 3$. (For matters of simplicity we state the result only outside these critical values.) Indeed, for $r\geq 2$ we have \[ C(\b_c(r)\mp 0,r) =(r-1)^{\frac{2(r-1)}{r(r-2)}}\times \left\{ \begin{array}{ll} r& \cr r (r-1)^\frac{r-2}{r} & \end{array} \right. \] which jumps for $r\geq 3$. (For $r=2$ this expression has to be interpreted as the limit of the right hand side with $r\downarrow 2$.) %It is not difficult to see that %\begin{eqnarray} % \exp\Bigl( \frac{\tilde\b(r-1)u}{r}\Bigr) %+ (r-1)\exp\Bigl( %-\frac{\tilde\b \,u}{r}\Bigr)>r %\end{eqnarray} %whenever $u>0$. %\noindent{\bf Example: } Let us consider the case $q=4$, $s=2$, $r_1=3$, $r_2=1$. %Then the single site-kernel is uniquely described by the real function %\begin{eqnarray} %&\lim_{N\uparrow \infty}\frac{Q^N_{4, \beta, (3,1)} %\bigl(1\bigl | (n_1,1-n_1) \bigr)}{ %Q^N_{4, \beta, (3,1)} %\bigl(2\bigl | (n_1,1-n_1) \bigr) %}=\exp\Bigl( \b\Bigl(\frac{4 n_1}{3}-1\Bigr) \Bigr)\cr %&\quad \times \left\{ %\begin{array}{ll} % 3,& \hbox{if}\,\, \b n_1<\b_c(3) %\cr % \exp\Bigl( \frac{2\b n_1 u(\b n_1,3)}{3}\Bigr) %+ 2 \exp\Bigl( %-\frac{\b n_1 \,u(\b n_1,3)}{3}\Bigr) %,& \hbox{if} \,\, \b n_1>\b_c(3) %\end{array}\right. %\end{eqnarray} \medskip\noindent The reader should notice the following: First of all we have shown the pointwise existence of the limit \[ (n_l)_{1\leq l\leq s} \mapsto \lim_{N\uparrow \infty}Q^N_{q, \beta, (r_1, \ldots, r_s)} \bigl(k\bigl | (n_l)_{1\leq l\leq s} \bigr) \, . \] The notion of ``continuity of limiting conditional probabilities'' that was introduced in Theorem \ref{thm:main_result_on_complete_graph} has the precise meaning of continuity of the right hand side as a function on the closed set $\PP$ of $s$-dimensional probability vectors with respect to the ordinary Euclidean topology. From the explicit limiting formula given in the theorem and the well-known knowledge of the jumps of the order parameter the proof of the first three parts of our main theorem \ref{thm:main_result_on_complete_graph} is now immediate. %Next we see that, in order to produce this discontinuity %the value of some $n_{l}$ must be sufficiently large. %From this parts (i), (ii), (iii) of Theorem %\ref{thm:main_result_on_complete_graph} follow. %\begin{thm} \label{thm:Gibbs-nonGibbs-meanfield} %\noindent {\bf (i)} Suppose that $r_i\leq 2$ for all $i=1,\dots,s$. %Then the function %is a continuous function on the set, for all $\b$. %\medskip %\noindent Assume that $r_i\geq 3$ for some $i$ and put %$r_*:=\min\{ r\geq 3, r=r_i \hbox{ for some }i=1,\dots,s\}$. %Denote by $\b_c(r)$ the inverse critical temperature of %the $r$-state Potts model. Then the following holds. %\noindent{\bf (ii)} Then %it is continuous for all %$\b<\b_c(r_*)$ %\noindent{\bf (iii)} Then %it is not continuous for all $\b\geq \b_c(r_*)$. %\end{thm} \medskip\noindent {\bf Proof of Theorem {\ref{thm:main_result_on_complete_graph} (i),(ii),(iii):}} The points of discontinuity are precisely given by the values $n_k=\frac{\b_c(r_k)}{\b}$ for those $k$ with $r_k\geq 3$ for which $\frac{\b_c(r_k)}{\b}<1$. So (i) is immediate. To see (ii) and (iii) we use that $\b_c(r)$ is an increasing function of $r$. $\Cox$ \subsection{Typicality of continuity points -- ``almost sure Gibbsianness''} What can be said about the measure of the discontinuity points? We will answer this question now and prove the remaining part (iv) of Theorem \ref{thm:main_result_on_complete_graph}. To start with, from Theorem \ref{thm:Empirical measures} follows trivially by ``contraction'' that the typical values of the order parameter in the fuzzy model are as follows. (Recall that $e_{l}$ is the unit vector in the $l$'th coordinate direction of $\R^{s}$.) \begin{cor} \label{cor:2.15} We have \begin{eqnarray*} \lefteqn{\lim_{N\uparrow\infty} \mu^N_{q, \beta, (r_1, \ldots, r_s)}\Bigl( \frac{1}{N}\sum_{x=1}^N (I_{\{Y(x)=1\}},\dots, I_{\{Y(x)=s\}}) \in \cdot \Bigr)} \\ &=& \left\{ \begin{array}{ll} \d_{\frac{1}{q}(r_{1},r_{2},\dots,r_{s})}& \,\hbox{if}\,\, \b<\b_c(q) \cr \l_0(q)\d_{\frac{1}{q}(r_{1},r_{2},\dots,r_{s})}+ \sum_{l=1}^{s} \frac{(1-\l_0(q)) r_{l}}{q} \d_{u(\b,q) e_{l}+\frac{1-u(\b,q)}{q}(r_{1},r_{2},\dots,r_{s})} & \,\hbox{if} \,\, \b=\b_c(q)\cr \sum_{l=1}^{s} \frac{r_{l}}{q} \d_{u(\b,q) e_{l}+\frac{1-u(\b,q)}{q}(r_{1},r_{2},\dots,r_{s})} & \,\hbox{if} \,\, \b>\b_c(q) \, . \end{array}\right. \end{eqnarray*} \end{cor} In other words, the values for the fuzzy densities $n_{l}$ that occur with non-zero probability are: The values $r_l /q$ in the high-temperature regime (including the critical point) and the two values \[ n^{+}(\b,q,r_{l}) \equiv u(q,\b)+\frac{1-u(q,\b)}{q} r_{l} \] and \[ n^{-}(\b,q,r_{l}) \equiv \frac{1-u(q,\b)}{q} r_{l} \quad \Bigl( \leq n^{+}_{l}(\b,q,r_{l})\Bigl) \] in the low temperature regime (including the critical point). Now, the non-trivial question is: Can it happen that these values coincide with the points of discontinuity of the limiting conditional probability, for certain choices of the parameter? The following proposition tells us that this can never be the case, and so the points of discontinuity are always atypical. This immediately proves (iv) of Theorem \ref{thm:main_result_on_complete_graph}. As we will see the proof of the proposition is elementary but slightly tricky; it makes use of specific properties of the solution of the mean-field equation. In that sense it is the most difficult part of our analysis of the mean field fuzzy Potts model. \begin{prop} \label{prop:300} Assume that $q>r\geq 2$. \begin{description} \item{\bf (i)} For the high-temperature range $\b\leq \b_{c}(q)$ we have \[ \frac{r}{q} < \frac{\b_{c}(r)}{\b} \, . \] \item{\bf (ii)} For the low-temperature range $\b\geq \b_{c}(q)$ we have that \[ n^{-}(\b,q,r)< \frac{\b_{c}(r)}{\b} r$. This in turn is implied by the fact that $\frac{\b_{c}(q)}{q}$ is decreasing in $q$. It is obvious that this holds for large enough $q$, by the explicit expression for $\b_{c}(q)$. It is elementary to verify that it holds in fact for any $q\geq 2$. Next we prove (ii). We show first the right inequality %The claim (i) says %$u(q,\b)+\frac{1-u(q,\b)}{q} r> \frac{\b_{c}(r)}{\b}$ which is equivalent to \[ u(q,\b)> \frac{q }{q-r}\frac{\b_{c}(r)}{\b}-\frac{r}{q-r} \, . \] By Theorem \ref{thm:Empirical measures} the order parameter $u(q,\b)$ is an increasing function in $\b$. The right hand side is decreasing in $\b$. So it suffices to prove the inequality for $\b=\b_c(q)$. Using $u(q,\b_c(q))=\frac{q-2}{q-1}$ this can be put equivalently as \begin{equation}\label{eq:3.4} \b_{c}(r)< %\b_c(q)\frac{q(q-2)+r}{q(q-1)} \b_c(q)\left(1-\frac{q-r}{q(q-1)}\right) \, . \end{equation} We will use now the elementary property that \begin{eqnarray} \label{eq:3.5} \b_c(q) 2 \, . \end{eqnarray} This implies also that $\b_c(q)$ is concave because \[ \b''_c(q)=\frac{-2 q (q-2)+ 4 (q-1)\log(q-1)}{(q-2)^3 (q-1)} \] and the denominator is negative, by the last inequality. In order to show (\ref{eq:3.4}) we note, by concavity that \begin{eqnarray}\label{eq:3.7} \b_{c}(r)\leq \b_c(q)+\b'_c(q) (r-q) \end{eqnarray} and show that the right hand side of (\ref{eq:3.7}) is strictly bounded from above by the right hand side of (\ref{eq:3.4}). But the latter statement is equivalent to \[ \b'_{c}(q)> \b_c(q)\frac{1}{q(q-1)} \, . \] Computing the logarithmic derivative $\frac{\b'_{c}(q)}{\b_c(q)}$ we see that this is equivalent to \[ \frac{1}{q-1}-\frac{1}{q-2}+\frac{1}{(q-1)\log(q-1)}> \frac{1}{q(q-1)} \, . \] This inequality in turn reduces after trivial computation to the statement (\ref{eq:3.5}) and this concludes the proof of the right inequality of (i). Let us come to the proof of the left inequality of (ii). The claim says $\frac{1-u(q,\b)}{q} r< \frac{\b_{c}(r)}{\b}$. Using the mean-field equation we may write \[ 1-u(q,\b) =\frac{q}{e^{+\b u(q,\b)}+q-1} \, . \] So the claim is equivalent to \[ \b \frac{r}{\b_c(r)}