Christian R. Scullard

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 Wiermans 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.

Density-functional calculations of transport properties in the nondegenerate limit and the role of electron-electron scattering

We compute electrical and thermal conductivities of hydrogen plasmas in the nondegenerate regime using Kohn-Sham density functional theory (DFT) and an application of the Kubo-Greenwood response formula, and demonstrate that for thermal conductivity, the mean-field treatment of the electron-electron (e-e) interaction therein is insufficient to reproduce the weak-coupling limit obtained by plasma kinetic theories. An explicit e-e scattering correction to the DFT is posited by appealing to Matthiessens Rule and the results of our computations of conductivities with the quantum Lenard-Balescu (QLB) equation. Further motivation of our correction is provided by an argument arising from the Zubarev quantum kinetic theory approach. Significant emphasis is placed on our efforts to produce properly converged results for plasma transport using Kohn-Sham DFT, so that an accurate assessment of the importance and efficacy of our e-e scattering corrections to the thermal conductivity can be made.

Critical surfaces for general inhomogeneous bond percolation problems

We present a method of general applicability for finding exact values or accurate approximations of bond percolation thresholds for a wide class of lattices. To every lattice we systematically 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: pc = 0.524 430..., (3, 122): pc = 0.740 423..., (33, 42): pc = 0.419 615..., (3, 4, 6, 4): pc = 0.524 821..., (4, 82): pc = 0.676 835..., (32, 4, 3, 4): pc = 0.414 120... . The results are generally within 10 − 5 of numerical estimates. For the inhomogeneous checkerboard and bow-tie lattices, errors in the formulae (if they are not exact) are less than 10 − 6.

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 which provides an approximation to the inhomogeneous kagomé and bond problems, and the other which gives estimates of for the homogeneous kagomé (0.524 408 8...) and (3, 12(2)) (0.740 421 2...) problems that, respectively, agree with numerical results to five and six significant figures.

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.

The computation of bond percolation critical polynomials by the deletion–contraction algorithm

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 extended to any periodic lattice. The polynomial is computed on a finite subgraph, called the base, of an infinite lattice. For any problem with exactly known solution, the prediction of the bond threshold is always correct for any base containing an arbitrary number of unit cells. For unsolved problems, the polynomial is referred to as the generalized critical polynomial and provides an approximation that becomes more accurate with increasing number of bonds in the base, appearing to approach the exact answer. The polynomials are computed using the deletion–contraction algorithm, which quickly becomes intractable by hand for more than about 18 bonds. Here, I present generalized critical polynomials calculated with a computer program for bases of up to 36 bonds for all the unsolved Archimedean lattices, except the kagome lattice, which was considered in an earlier work. The polynomial estimates are generally within 10−5–10−7 of the numerical values, but the prediction for the (4,82) lattice, though not exact, is not ruled out by simulations.

Phase diagram of the triangular-lattice Potts antiferromagnet

We study the phase diagram of the triangular-lattice

Magnesium isotope fractionation by chemical diffusion in natural settings and in laboratory analogues

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Exact site percolation thresholds using a site-to-bond transformation and the star-triangle transformation.

-state Potts model in the real

Analytic expressions for electron-ion temperature equilibration rates from the Lenard-Balescu equation

(Q,v)