Building upon his previous work with Scullard (see below), Jesper Lykke Jacobsen shows how to reformulate their graph polynomial method used access to the critical temperature of the q-state Potts model on a general two-dimensional lattice, into an eigenvalue method.

Three examples of connectivity states for n = 4. Loops are shown as red solid lines. The corresponding FK clusters live in the areas shaded in grey, while the dual FK clusters live in the white areas.

The numerous advantages of this reformulation are discussed and demonstrated by applying the method to three significant, unsolved problems:

1. Bond percolation on the kagome lattice.
2. Site percolation on the square lattice.
3. Self-avoiding polygons (SAPs) on the square lattice

This paper is one from the JPhysA Special Issue:  Exactly Solved Models and Beyond produced in honour of R J Baxter’s 75th birthday. It is a further reflection of the ongoing influence of Baxter’s work.

Click here to read the paper

Jesper Lykke Jacobsen works within the Laboratory of Theoretical Physics at École Normale Supérieure a member of Paris Sciences et Lettres Research University. His research interests include statistical physics, disordered systems, conformal field theory and exactly solvable models.

Read more from the authors:

High-precision percolation thresholds and Potts-model critical manifolds from graph polynomials 

Transfer matrix computation of critical polynomials for two-dimensional Potts models

Transfer matrix computation of generalized critical polynomials in percolation 

Critical manifold of kagome-lattice Potts model

This work is licensed under a Creative Commons Attribution 3.0 Unported License. Image: Jesper Lykke Jacobsen 2015 J. Phys. A: Math. Theor. 48 454003 copyright IOP Publishing Ltd 2015

Categories: Journal of Physics A: Mathematical and Theoretical

Tags: Computational physics, Image of the week, Mathematical physics, Special Issue, Statistical physics