The covariance matrix of the potts model: A random cluster analysis

Abstract

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.