Diameter of Random Cluster Models

The diameter of critical clusters of the $q$-state Potts model in the random cluster representation is measured in numerical simulations. The diameter of a graph is the maximum over all pairs of connected sites of the minimum path length between the sites. Although the diameter of Fortuin-Kasteleyn clusters is not a thermodynamic quantity, it is expected to display power law critical behavior since the size of the largest cluster diverges at the critical point. The Swendsen-Wang algorithm is used for both for equilibrating the spin model and for identifying clusters. An efficient algorithm is employed to measure the diameter. The exponent characterizing the divergence of the diameter is obtained for $q=1$,2,3 and 4. The relation to other critical exponents is discussed. This work is supported by NSF (DMR-0242402).