arXiv subjects

Christian R. Scullard

Publications and source records attributed to Christian R. Scullard.

23 records · Page 2Linked to original sources

arXiv2011

Polynomial sequences for bond percolation critical thresholds

In this paper, I compute the inhomogeneous (multi-probability) bond critical surfaces for the (4,6,12) and (3^4,6) lattices using the linearity approximation described in (Scullard and Ziff, J. Stat. Mech. P03021), implemented as a branching process of lattices. I find the estimates for the bond percolation thresholds, p_c(4,6,12)=0.69377849... and p_c(3^4,6)=0.43437077..., compared with Parviainen's numerical results of p_c \approx 0.69373383 and p_c \approx 0.43430621 . These deviations are of the order 10^{-5}, as is standard for this method, although they are outside Parviainen's typical standard error of 10^{-7}. Deriving thresholds in this way for a given lattice leads to a polynomial with integer coefficients, the root in [0,1] of which gives the estimate for the bond threshold. I show how the method can be refined, leading to a sequence of higher order polynomials making predictions that likely converge to the exact answer. Finally, I discuss how this fact hints that for certain graphs, such as the kagome lattice, the exact bond threshold may not be the root of any polynomial with integer coefficients.

cond-mat.dis-nn↗

arXiv2010

Critical surfaces for general inhomogeneous bond percolation problems

We present a method of general applicability for finding exact or accurate approximations to bond percolation thresholds for a wide class of lattices. To every lattice we sytematically 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, in which all vertices are equivalent, but this restriction is not crucial. Some results we find are kagome: $p_c=0.524430...$, $(3,12^2): p_c=0.740423...$, $(3^3,4^2): p_c=0.419615...$, $(3,4,6,4):p_c=0.524821...$, $(4,8^2):p_c=0.676835...$, $(3^2,4,3,4)$: $p_c=0.414120...$ . The results are generally within $10^{-5}$ of numerical estimates. For the inhomogeneous checkerboard and bowtie lattices, errors in the formulas (if they are not exact) are less than $10^{-6}$.

cond-mat.stat-mech↗

arXiv2008

Critical surfaces for general bond percolation problems

We present a general method for predicting bond percolation thresholds and critical surfaces for a broad class of two-dimensional periodic lattices, reproducing many known exact results and providing excellent approximations for several unsolved lattices. For the checkerboard and inhomogeneous bow-tie lattices, the method yields predictions that agree with numerical measurements to more than six figures, and are possibly exact.

cond-mat.dis-nn↗

arXiv2006

Exact bond percolation thresholds in two dimensions

Recent work in percolation has led to exact solutions for the site and bond critical thresholds of many new lattices. Here we show how these results can be extended to other classes of graphs, significantly increasing the number and variety of solved problems. Any graph that can be decomposed into a certain arrangement of triangles, which we call self-dual, gives a class of lattices whose percolation thresholds can be found exactly by a recently introduced triangle-triangle transformation. We use this method to generalize Wierman's solution of the bow-tie lattice to yield several new solutions. We also give another example of a self-dual arrangement of triangles that leads to a further class of solvable problems. There are certainly many more such classes.

cond-mat.dis-nn↗

arXiv2006

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

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

cond-mat.dis-nn↗