# Percolation critical polynomial as a graph invariant.

@article{Scullard2011PercolationCP, title={Percolation critical polynomial as a graph invariant.}, author={Christian R Scullard}, journal={Physical review. E, Statistical, nonlinear, and soft matter physics}, year={2011}, volume={86 4 Pt 1}, pages={ 041131 } }

Every lattice for which the bond percolation critical probability can be found exactly possesses a critical polynomial, with the root in [0,1] providing the threshold. Recent work has demonstrated that this polynomial may be generalized through a definition that can be applied on any periodic lattice. The polynomial depends on the lattice and on its decomposition into identical finite subgraphs, but once these are specified, the polynomial is essentially unique. On lattices for which the exact…

## 12 Citations

### 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

- 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…

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

- Mathematics
- 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…

### 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.…

### Critical percolation on the kagome hypergraph

- PhysicsJournal of Physics A: Mathematical and Theoretical
- 2021

We study the percolation critical surface of the kagome lattice in which each triangle is allowed an arbitrary connectivity. Using the method of critical polynomials, we find points along this…

### Bond percolation thresholds on Archimedean lattices from critical polynomial roots

- 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…

### 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…

### 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…

### On the growth constant for square-lattice self-avoiding walks

- Mathematics
- 2016

The growth constant for two-dimensional self-avoiding walks on the honeycomb lattice was conjectured by Nienhuis in 1982, and since that time the corresponding results for the square and triangular…

### New bounds for the site percolation threshold of the hexagonal lattice

- Computer ScienceJournal of Physics A: Mathematical and Theoretical
- 2022

The site percolation threshold of the hexagonal lattice satisfies 0.656 246 < p c < 0.739 695, and this bound is obtained by using the substitution method to compare the hexagon lattice site model to an exactly-solved two-parameter site perColation model on the martini lattice.

## References

### C: Solid State Phys

- 15, L757
- 1982