# Critical surfaces for general inhomogeneous bond percolation problems

@article{Scullard2009CriticalSF, title={Critical surfaces for general inhomogeneous bond percolation problems}, author={Christian R Scullard and Robert M. Ziff}, journal={Journal of Statistical Mechanics: Theory and Experiment}, year={2009}, volume={2010}, pages={P03021} }

We present a method of general applicability for finding exact values or accurate approximations of bond percolation thresholds for a wide class of lattices. To every lattice we systematically associate a polynomial, the root of which in [0, 1] is the conjectured critical point. The method makes the correct prediction for every exactly solved problem, and comparison with numerical results shows that it is very close, but not exact, for many others. We focus primarily on the Archimedean lattices…

## 16 Citations

### 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…

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

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- 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…

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- PhysicsPhysical Review Research
- 2020

We present percolation thresholds calculated numerically with the eigenvalue formulation of the method of critical polynomials; developed in the last few years, it has already proven to be orders of…

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

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- 2014

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…

### Potts-model critical manifolds revisited

- Physics
- 2015

We compute critical polynomials for the q-state Potts model on the Archimedean lattices, using a parallel implementation of the algorithm of Jacobsen (2014 J. Phys. A: Math. Theor 47 135001) that…

### On bond percolation threshold bounds for Archimedean lattices with degree three

- Mathematics
- 2017

Improved bounds are proved for bond percolation thresholds for certain Archimedean lattices using the substitution method with new comparisons between models and more efficient computational…

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

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- 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…

### How Inhomogeneous Site Percolation Works on Bethe Lattices: Theory and Application

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- 2016

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### Tight bounds for the bond percolation threshold of the (3, 122) lattice

- Computer Science
- 2016

Improved mathematically rigorous upper and lower bounds for the bond percolation threshold are established using the substitution method, in which stochastic ordering is checked using a network flow algorithm.

## References

SHOWING 1-10 OF 44 REFERENCES

### 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.…

### Exact Critical Percolation Probabilities for Site and Bond Problems in Two Dimensions

- Mathematics
- 1964

An exact method for determining the critical percolation probability, pc, for a number of two‐dimensional site and bond problems is described. For the site problem on the plane triangular lattice pc…

### Predictions of bond percolation thresholds for the kagomé and Archimedean (3, 12(2)) lattices.

- MathematicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2006

Here we show how the recent exact determination of the bond percolation threshold for the martini lattice can be used to provide approximations to the unsolved kagomé and (3, 12(2)) lattices. We…

### Percolation thresholds on two-dimensional Voronoi networks and Delaunay triangulations.

- Computer SciencePhysical review. E, Statistical, nonlinear, and soft matter physics
- 2009

The results rule out the conjecture by Hsu and Huang that the bond thresholds are 2/3 and 1/3, respectively, but support the conjecture of Wierman that, for fully triangulated lattices other than the regular triangular lattice, the bond threshold is less than 2 sin pi/18 approximately 0.3473.

### Exact site percolation thresholds using a site-to-bond transformation and the star-triangle transformation.

- MathematicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2006

A correlated bond problem on the hexagonal lattice is solved by use of the star-triangle transformation and the site problem is solved, by a particular choice of correlations derived from a site-to-bond transformation, on the martini lattice.

### Phase diagram of anisotropic planar Potts ferromagnets: a new conjecture

- Physics
- 1982

The exact phase diagram of the nearest-neighbour q-state Potts ferromagnet in the fully anisotropic 3-12 lattice is conjectured through a star-triangle transformation. It recovers all the available…

### Universal condition for critical percolation thresholds of kagomé-like lattices.

- Computer SciencePhysical review. E, Statistical, nonlinear, and soft matter physics
- 2009

Lattices that can be represented in a kagomé-like form are shown to satisfy a universal percolation criticality condition, expressed as a relation between P3 , the probability that all three vertices…

### Using Symmetry to Improve Percolation Threshold Bounds

- Computer ScienceCombinatorics, Probability and Computing
- 2005

It is shown that symmetry, represented by a graph's automorphism group, can be used to greatly reduce the computational work for the substitution method, resulting in tighter bounds on the percolation threshold $p_c$.

### Critical percolation in high dimensions.

- Computer SciencePhysical review. E, Statistical, nonlinear, and soft matter physics
- 2003

Monte Carlo estimates for site and bond percolation thresholds in simple hypercubic lattices with 4-13 dimensions are presented and a scaling law for finite cluster size corrections is proposed.

### Critical Percolation of Free Product of Groups

- MathematicsInt. J. Algebra Comput.
- 2008

It is shown that the critical probabilities of the free product of these approximations converge to the critical probability of G1 * G2 * ⋯ * Gn and the speed of convergence is exponential, which means that for residually finite groups, for example, one can restrict oneself to the case when each free factor is finite.