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We report numerical simulations of two-dimensional $q$-state Potts modelswith emphasis on a new quantity for the computation of spatial correlationlengths. This quantity is the cluster-diameter distribution function$G_{diam}(x)$, which measures the distribution of the diameter ofstochastically defined cluster. Theoretically it is predicted to fall offexponentially for large diameter $x$, $G_{diam} \propto \exp(-x/\xi)$, where$\xi$ is the correlation length as usually defined through the large-distancebehavior of two-point correlation functions. The results of our extensive MonteCarlo study in the disordered phase of the models with $q=10$, 15, and $20$ onlarge square lattices of size $300 \times 300$, $120 \times 120$, and $80\times 80$, respectively, clearly confirm the theoretically predicted behavior.Moreover, using this observable we are able to verify an exact formula for thecorrelation length $\xi_d(\beta_t)$ in the disordered phase at the first-ordertransition point $\beta_t$ with an accuracy of about $1%-2%$ for all consideredvalues of $q$. This is a considerable improvement over estimates derived fromthe large-distance behavior of standard (projected) two-point correlationfunctions, which are also discussed for comparison.

Monte Carlo study of cluster-diameter distribution: An observable to estimate correlation lengths