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

## TL;DR

This paper introduces the cluster-diameter distribution function as a new observable to accurately estimate correlation lengths in 2D Potts models, confirmed through extensive Monte Carlo simulations.

## Contribution

The study demonstrates that the cluster-diameter distribution function provides a precise and reliable method to measure correlation lengths, improving upon traditional two-point correlation function estimates.

## Key findings

- Confirmed exponential decay of cluster-diameter distribution in simulations.
- Verified an exact formula for correlation length at the first-order transition point.
- Achieved 1-2% accuracy in correlation length estimates using the new observable.

## 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 $1%-2%$ 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.