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

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.

Automated summary

We propose a method for computing transition points of the random cluster model using a generalization of the Newman-Ziff algorithm. The method is easy to implement and works for real cluster weight q > 0. It allows for obtaining results for an arbitrary number of values of q within a single simulation. However, the accuracy is reduced for q > 1 on large lattices. Nonetheless, accurate estimates of critical points in two dimensions can be obtained by sampling the critical polynomial, as demonstrated in this study on the square lattice and the unsolved non-planar square matching lattice.