Critical points of the random cluster model with Newman-Ziff sampling

TL;DR

This paper introduces a generalized Newman-Ziff algorithm for efficiently computing critical points in the random cluster model across various cluster weights, enabling accurate estimates even on small lattices and for non-integer q values.

Contribution

The authors extend the Newman-Ziff algorithm to the random cluster model, allowing simultaneous computation of critical points for multiple q values with improved efficiency and accuracy.

Findings

Accurate critical point estimates for non-integer q on square lattices.

Method works efficiently with small lattice sizes.

Results obtained for non-planar square matching lattice, previously difficult to analyze.

Abstract

We present a method for computing transition points of the random cluster model using a generalization of the Newman-Ziff algorithm, a celebrated technique in numerical percolation, to the random cluster model. The new method is straightforward to implement and works for real cluster weight $q>0$. Furthermore, results for an arbitrary number of values of $q$ can be found at once within a single simulation. Because the algorithm used to sweep through bond configurations is identical to that of Newman and Ziff, which was conceived for percolation, the method loses accuracy for large lattices when $q>1$. However, by sampling the critical polynomial, accurate estimates of critical points in two dimensions can be found using relatively small lattice sizes, which we demonstrate here by computing critical points for non-integer values of $q$ on the square lattice, to compare with the exact solution, and on the unsolved non-planar square matching lattice. The latter results would be much more difficult to obtain using other techniques.