THE COVARIANCE-MATRIX OF THE POTTS-MODEL - A RANDOM CLUSTER-ANALYSIS

We consider the covariance matrix, G(mn) = q(2)[delta(sigma(x),m);delt a(sigma(y),n)], of the d-dimensional q-states Potts model, rewriting i t in the random cluster representation of Fortuin and Kasteleyn. In an y of the q ordered phases, we identify the eigenvalues of this matrix both in terms of representations of the unbroken symmetry group of the model and in terms of random cluster connectivities and covariances, thereby attributing algebraic significance to these stochastic geometr ic quantities. We also show that the correlation length corresponding to the decay rate of one of the eigenvalues is the same as the inverse decay rate of the diameter of finite clusers. For dimension d = 2, we show that this correlation length and the correlation length of the t wo-point function with free boundary conditions at the corresponding d ual temperature are equal up to a factor of two. For systems with firs t-order transitions, this relation helps to resolve certain inconsiste ncies between recent exact and numerical work on correlation lengths a t the self-dual point beta(0). For systems with second order transitio ns, this relation implies the equality of the correlation length expon ents from above and below threshold, as well as an amplitude ratio of two. In the course of proving the above results, we establish several properties of independent interest, including left continuity of the i nverse correlation length with free boundary conditions and upper semi continuity of the decay rate for finite clusters in all dimensions, an d left continuity of the two-dimensional free boundary condition perco lation probability at beta(0). We also introduce DLR equations for the random cluster model and use them to establish ergodicity of the free measure. In order to prove these results, we introduce a new class of events which we call decoupling events and two inequalities for these events. The first is similar to the FKG inequality, but holds for eve nts which are neither increasing nor decreasing; the second is similar to the van den Berg-Kesten inequality in standard percolation. Both i nequalities hold for an arbitrary FKG measure.