Exact bond percolation thresholds in two dimensions

Abstract

Abstract. 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.

Pseudo-random-number generators and the square site percolation threshold

“…A parametrised fit of this expression to the data of table III is shown in figure 4 and produces (9)) to experimental data (p h (L) of table III) for the linear combination critical point estimator. (7), and (b+αc) = 0.3 (9). The threshold result is in good agreement with those obtained from the median-p estimator data.…”

“…Yet to date, no analytical expression has been found for the numerical value of p c . The square site lattice lacks the symmetry that has allowed exact solutions on other topologies [2,4,5,6,7,8,9]. So long as the problem remains intractable, statistical estimates from Monte Carlo studies can, at least, offer approximate values.…”

Linear independence of nearest-neighbor valence bond states in several two-dimensional lattices

“…In Section II B we report that this method can be applied without much alteration to the honeycomb lattice, the star lattice, the square-octagon lattice, the squagome lattice, two types of pentagonal lattices (studied in a magnetic context, e.g., in Refs. 29 and 22), three types of "martini" lattices, 30 and two types of archimedean lattices. In Section II C, we apply the same method to fulleren-type lattices.…”

Percolation Theory

“…Their predictions were later confirmed in 1980-1982, showing p c = 1/2 for the square lattice bond model, p c = 2 sin (B/18) ≈ 0.347296 for the triangular lattice bond model and p c = 1 − 2 sin (B/18) ≈ 0.652704 for the hexagonal lattice bond model, and p c = 1/2 for the triangular lattice site model and the site model on any full-triangulated periodic lattice. More recent research produced an approach for constructing an infinite class of graphs for which bond percolation thresholds may be exactly determined [22,23] , and [24] . The solutions for bond models depend crucially on graph duality.…”