Diameter of Random Clusters in Potts Models

We report measurements of cluster diameter -- the maximum over all pairs of connected vertices of the minimum path length between the vertices -- in numerical simulations of random clusters in q-state Potts models in two and three dimensions. Although the diameter is not a thermodynamic quantity, it is expected to display critical behavior for Potts models models as the size of the largest cluster diverges at the critical point. We have developed an efficient algorithm for measuring the diameter, and have obtained results using the Swendsen-Wang algorithm both for equilibrating the model and for identifying clusters.