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

@article{Jacobsen2014HighprecisionPT, title={High-precision percolation thresholds and Potts-model critical manifolds from graph polynomials}, author={Jesper Lykke Jacobsen}, journal={Journal of Physics A: Mathematical and Theoretical}, year={2014}, volume={47} }

The critical curves of the q-state Potts model can be determined exactly for regular two-dimensional lattices G that are of the three-terminal type. This comprises the square, triangular, hexagonal and bow–tie lattices. Jacobsen and Scullard have defined a graph polynomial PB(q, v) that gives access to the critical manifold for general lattices. It depends on a finite repeating part of the lattice, called the basis B, and its real roots in the temperature variable v = eK − 1 provide…

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## References

SHOWING 1-10 OF 67 REFERENCES

### Critical manifold of the kagome-lattice Potts model

- Mathematics
- 2012

Any two-dimensional infinite regular lattice G can be produced by tiling the plane with a finite subgraph B⊆G; we call B a basis of G. We introduce a two-parameter graph polynomial PB(q, v) that…

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

- Mathematics
- 2013

In our previous work [1] we have shown that critical manifolds of the q-state Potts model can be studied by means of a graph polynomial PB(q, v), henceforth referred to as the critical polynomial.…

### Transfer matrix computation of generalized critical polynomials in percolation

- Mathematics
- 2012

Percolation thresholds have recently been studied by means of a graph polynomial PB(p), henceforth referred to as the critical polynomial, that may be defined on any periodic lattice. The polynomial…

### Critical frontier of the Potts and percolation models on triangular-type and kagome-type lattices. II. Numerical analysis.

- MathematicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2010

Numerical determination of critical properties such as conformal anomaly and magnetic correlation length verifies that the universality principle holds and infers that the homogeneity assumption determines critical frontiers with an accuracy of 5 decimal places or higher.

### Percolation critical polynomial as a graph invariant.

- MathematicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2012

This paper shows how the generalized critical polynomial can be viewed as a graph invariant, similar to the TuttePolynomial, and allows calculation on a computer for the kagome lattice using subgraphs of up to 36 bonds.

### Polynomial sequences for bond percolation critical thresholds

- Mathematics
- 2011

In this paper, I compute the inhomogeneous (multi-probability) bond critical surfaces for the (4, 6, 12) and (34, 6) lattices using the linearity approximation described in Scullard and Ziff (2010 J.…

### The computation of bond percolation critical polynomials by the deletion–contraction algorithm

- Computer Science
- 2012

Although every exactly known bond percolation critical threshold is the root in [0,1] of a lattice-dependent polynomial, it has recently been shown that the notion of a critical polynomial can be…

### Spanning Forests and the q-State Potts Model in the Limit q →0

- Physics
- 2005

We study the q-state Potts model with nearest-neighbor coupling v=eβJ−1 in the limit q,v → 0 with the ratio w = v/q held fixed. Combinatorially, this limit gives rise to the generating polynomial of…

### Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. III. Triangular-Lattice Chromatic Polynomial

- Mathematics
- 2003

We study the chromatic polynomial PG(q) for m×n triangular-lattice strips of widths m≤12P,9F (with periodic or free transverse boundary conditions, respectively) and arbitrary lengths n (with free…