Monte Carlo Study of Cluster-Diameter Distribution: A New Observable to Estimate Correlation Lengths

We report numerical simulations of two-dimensional q-state Potts models with emphasis on a new quantity for the computation of spatial correlation lengths. This quantity is the cluster-diameter distribution function Gdiam(x), which measures the distribution of the diameter of stochastically defined cluster. Theoretically it is predicted to fall off exponentially for large diameter x, Gdiam ∝ exp(−x/ξ), where ξ is the correlation length as usually defined through the large-distance behavior of two-point correlation functions. The results of our extensive Monte Carlo study in the disordered phase of the models with q = 10, 15, and 20 on large square lattices of size 300× 300, 120× 120, and 80 × 80, respectively, clearly confirm the theoretically predicted behavior. Moreover, using this observable we are able to verify an exact formula for the correlation length ξd(βt) in the disordered phase at the first-order transition point βt with an accuracy of about 1%− 2% for all considered values of q. This is a considerable improvement over estimates derived from the large-distance behavior of standard (projected) two-point correlation functions, which are also discussed for comparison. PACS numbers: 05.50.+q, 75.10.Hk, 64.60.Cn, 11.15.Ha