[Paper Review] Transfer matrix computation of generalised critical polynomials in percolation

Christian R. Scullard, Jesper Lykke Jacobsen|arXiv (Cornell University)|Sep 7, 2012

Theoretical and Computational Physics3 references3 citations

TL;DR

This paper introduces a probabilistic transfer matrix method to compute generalized critical polynomials in percolation, enabling efficient and accurate calculation of percolation thresholds on 2D lattices. By redefining the critical polynomial via a probabilistic crossing probability on a finite basis, the method overcomes the computational limits of prior contraction-deletion approaches, achieving polynomial degrees up to 243 and yielding highly precise estimates for bond percolation thresholds on the (4,8²), kagome, and (3,12²) lattices.

ABSTRACT

Percolation thresholds have recently been studied by means of a graph polynomial $P_B(p)$, henceforth referred to as the critical polynomial, that may be defined on any periodic lattice. The polynomial depends on a finite subgraph $B$, called the basis, and the way in which the basis is tiled to form the lattice. The unique root of $P_B(p)$ in $[0,1]$ either gives the exact percolation threshold for the lattice, or provides an approximation that becomes more accurate with appropriately increasing size of $B$. Initially $P_B(p)$ was defined by a contraction-deletion identity, similar to that satisfied by the Tutte polynomial. Here, we give an alternative probabilistic definition of $P_B(p)$, which allows for much more efficient computations, by using the transfer matrix, than was previously possible with contraction-deletion. We present bond percolation polynomials for the $(4,8^2)$, kagome, and $(3,12^2)$ lattices for bases of up to respectively 96, 162, and 243 edges, much larger than the previous limit of 36 edges using contraction-deletion. We discuss in detail the role of the symmetries and the embedding of $B$. For the largest bases, we obtain the thresholds $p_c(4,8^2) = 0.676 803 329 ...$, $p_c(\mathrm{kagome}) = 0.524 404 998 ...$, $p_c(3,12^2) = 0.740 420 798 ...$, comparable to the best simulation results. We also show that the alternative definition of $P_B(p)$ can be applied to study site percolation problems.

Motivation & Objective

• To overcome the computational bottleneck of the contraction-deletion method for computing generalized critical polynomials in percolation.
• To develop a more efficient computational framework for estimating percolation thresholds on unsolved 2D lattices.
• To validate the conjecture that the root of the generalized critical polynomial in [0,1] converges to the exact percolation threshold as the basis size increases.
• To explore the role of basis symmetry and embedding in the accuracy of threshold predictions.
• To extend the method to site percolation and higher-dimensional generalizations, though the latter remains open.

Proposed method

• Reformulate the generalized critical polynomial $P_B(p)$ using a probabilistic definition based on the toroidal crossing probability of a finite basis $B$.
• Apply the transfer matrix method to efficiently compute the crossing probability for large bases, replacing the computationally expensive contraction-deletion recursion.
• Use the condition $P_{\text{cross}}(p) = P_{\text{nocross}}(p)$ to define the critical polynomial, where $P_{\text{cross}}$ is the probability of a crossing path and $P_{\text{nocross}}$ is the probability of no connection.
• Implement parallelized transfer matrix algorithms to handle large bases with up to 243 edges, significantly exceeding the previous 36-edge limit.
• Systematically vary the basis shape (square and hexagonal) and size to test convergence and universality of the threshold prediction.
• Generalize the method to site percolation by applying the same crossing probability condition to the covering lattice or with correlated sites.

Experimental results

Research questions

• RQ1 Can the generalized critical polynomial be computed efficiently beyond the 36-edge limit previously imposed by the contraction-deletion method?
• RQ2 Does the probabilistic transfer matrix method yield more accurate percolation threshold estimates than previous approaches for unsolved lattices?
• RQ3 Is the critical threshold prediction via $P_B(p)$ independent of the basis shape and embedding, as conjectured, when the basis size increases?
• RQ4 How does the ratio $\zeta(n)$ of boundary vertices to internal elements affect the convergence speed of the threshold estimate?
• RQ5 Can the same probabilistic definition of $P_B(p)$ be extended to site percolation and the $q$-state Potts model with comparable accuracy?

Key findings

• The transfer matrix method enables computation of generalized critical polynomials up to degree 243, far exceeding the prior 36-edge limit of contraction-deletion.
• For the (4,8²) lattice, the method yields $p_c = 0.676803329\cdots$, matching the best simulation results.
• For the kagome lattice, the threshold is estimated as $p_c = 0.524404998\cdots$, consistent with state-of-the-art numerical simulations.
• For the (3,12²) lattice, the result $p_c = 0.740420798\cdots$ is in excellent agreement with current numerical estimates.
• The method performs best when the ratio $\zeta(n)$ of boundary vertices to internal elements decreases rapidly with basis size, with hexagonal bases showing faster convergence than square bases.
• The results support the conjecture that the critical polynomial root converges to the exact threshold in the limit of large, appropriately shaped bases, regardless of embedding or aspect ratio.