[Paper Review] Precise predictions of bond percolation thresholds for the kagom\'e and $(3,12^2)$ lattices

Christian R. Scullard, Robert M. Ziff|arXiv (Cornell University)|Feb 17, 2006

Geometry and complex manifolds3 citations

TL;DR

This paper uses the exact bond percolation threshold of the martini lattice to derive highly accurate approximations for the unsolved kagome and (3,12²) lattices. It introduces two methods: one for inhomogeneous systems and another yielding precise estimates of $p_c = 0.5244088$ for the homogeneous kagome and $p_c = 0.7404212$ for the (3,12²) lattice, matching numerical results to five and six significant figures respectively.

ABSTRACT

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 kagome and (3,12^2) lattices. We present two different methods, one of which provides an approximation to the inhomogeneous kagome and (3,12^2) bond problems, and the other gives estimates of $p_c$ for the homogeneous kagome (0.5244088...) and (3,12^2) (0.7404212...) problems that respectively agree with numerical results to five and six significant figures.

Motivation & Objective

• To address the lack of exact solutions for bond percolation thresholds in the kagome and (3,12²) lattices.
• To leverage the recently determined exact threshold of the martini lattice as a foundation for approximating these unsolved systems.
• To develop methods that yield highly precise estimates of $p_c$ for both homogeneous and inhomogeneous versions of the kagome and (3,12²) lattices.
• To achieve agreement with numerical simulations to high precision, validating the accuracy of the proposed approximations.

Proposed method

• Utilize the exact bond percolation threshold of the martini lattice as a reference point for deriving approximations.
• Apply a mapping or transformation technique to extend the martini lattice solution to the kagome and (3,12²) lattices.
• Develop a method for approximating inhomogeneous bond percolation problems on the kagome and (3,12²) lattices by exploiting structural similarities.
• Implement a second, distinct method focused on homogeneous lattices to compute precise values of $p_c$.
• Use iterative or numerical refinement techniques to improve convergence and accuracy of the estimated thresholds.
• Validate the results by comparing the predicted $p_c$ values with existing numerical simulations.

Experimental results

Research questions

• RQ1 Can the exact bond percolation threshold of the martini lattice be used to derive accurate approximations for the unsolved kagome and (3,12²) lattices?
• RQ2 What methodological approach enables precise estimation of $p_c$ for the homogeneous kagome lattice, yielding agreement with numerical results to five significant figures?
• RQ3 How can the framework be extended to handle inhomogeneous bond percolation problems on the kagome and (3,12²) lattices?
• RQ4 What is the achievable precision of the estimated $p_c$ for the (3,12²) lattice, and how does it compare to numerical simulations?
• RQ5 To what extent do the derived approximations reflect the true critical thresholds of these lattices?

Key findings

• The method yields a precise estimate of $p_c = 0.5244088$ for the homogeneous kagome lattice, matching numerical results to five significant figures.
• For the (3,12²) lattice, the method produces $p_c = 0.7404212$, agreeing with numerical simulations to six significant figures.
• An alternative approach provides approximations for the inhomogeneous bond percolation problems on both the kagome and (3,12²) lattices.
• The accuracy of the estimates is validated through comparison with numerical data, confirming high reliability.
• The results demonstrate that the martini lattice's exact solution enables highly accurate predictions for structurally related unsolved lattices.
• The framework offers a robust pathway to estimating critical thresholds in complex lattice systems where exact solutions remain unknown.