  We consider the covariance matrix $G^{mn}(x-y)$ of the d-dimensional q-states
Potts model, rewriting it in terms of the connectivity, the finite-cluster
connectivity and the infinite-cluster covariance in the random cluster repre-
sentation of Fortuin and Kasteleyn. In any of the $q$ ordered phases, we show
that the matrix $G^{mn}(x-y)$ has one tivial eigenvalue 0, one simple eigen-
value $G_{\wir}^{(1)}(x-y)$ and one ($q-2$)-fold degenerate eigenvalue
$G_{\wir}^{(2)}(x-y)$. Furthermore, we identify the eigenvalues both in terms
of representations of the unbroken symmetry group of the model, and in terms of
connectivities and cluster covariances, thereby attributing algebraic signifi-
cance to these stochastic geometric quantities. In addition to establishing the
existence of the correlation lengths $\xi_{\wir}^{(1)}$ and $\xi_{\wir}^{(2)}$
corresponding to $G_{\wir}^{(1)}(x-y)$ and $G_{\wir}^{(2)}(x-y)$, we show that
$\xi_{\wir}^{(1)}(\beta)\geq \xi_{\wir}^{(2)}(\beta)$ for all inverse tempera-
tures $\beta$. For dimension $d=2$ and $q \geq 1$, we establish a duality
relation between $\xi_{\wir}^{(2)}$ and $\xi_{\free}$, the correlation length
of the two-point function with free boundary conditions: We show
$\xi_{\wir}^{(2)}(\beta) = \frac{1}{2} \xi_{\free}(\beta^\ast)$ for all $\beta
\geq \beta_o$, where $\beta^\ast$ is the dual inverse temperature and $\beta_o$
is the self-dual point. In order to prove the above results, we introduce two
new inequalities. The first is similar to the FKG inequality, but holds for
events which are neither increasing nor decreasing, and replaces independence
in the standard percolation model; the second replaces the van den Berg -
Kesten inequality.
