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

Abstract

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 $G_{diam}(x)$, which measures the distribution of the diameter of stochastically defined cluster. Theoretically it is predicted to fall off exponentially for large diameter $x$, $G_{diam}\propto \exp (-x/\xi )$, where $\xi$ 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 $300\times 300$, $120\times 120$, and $80\times 80$, respectively, clearly confirm the theoretically predicted behavior. Moreover, using this observable we are able to verify an exact formula for the correlation length ${\xi }_d({\beta }_t)$ in the disordered phase at the first-order transition point ${\beta }_t$ with an accuracy of about $1$ 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.

Comments: 20 pages, LaTeX + 13 postscript figures. See also http://www.cond-mat.physik.uni-mainz.de/~janke/doc/home_janke.html