Universal Formulae for Percolation Thresholds

A power law is postulated for both site and bond percolation thresholds. The formula writes $p_c=p_0[(d-1)(q-1)]^{-a}d^{\ b}$, where $d$ is the space dimension and $q$ the coordination number. All thresholds up to $d\rightarrow \infty$ are found to belong to only three universality classes. For first two classes $b=0$ for site dilution while $b=a$ for bond dilution. The last one associated to high dimensions is characterized by $b=2a-1$ for both sites and bonds. Classes are defined by a set of value for $\{p_0; \ a\}$. Deviations from available numerical estimates at $d \leq 7$ are within $\pm 0.008$ and $\pm 0.0004$ for high dimensional hypercubic expansions at $d \geq 8$. The formula is found to be also valid for Ising critical temperatures.Comment: 11 pages, latex, 3 figures not include

Critical Percolation in High Dimensions

We present Monte Carlo estimates for site and bond percolation thresholds in simple hypercubic lattices with 4 to 13 dimensions. For d<6 they are preliminary, for d >= 6 they are between 20 to 10^4 times more precise than the best previous estimates. This was achieved by three ingredients: (i) simple and fast hashing which allowed us to simulate clusters of millions of sites on computers with less than 500 MB memory; (ii) a histogram method which allowed us to obtain information for several p values from a single simulation; and (iii) a new variance reduction technique which is especially efficient at high dimensions where it reduces error bars by a factor up to approximately 30 and more. Based on these data we propose a new scaling law for finite cluster size corrections.Comment: 5 pages including figures, RevTe

Counting Lattice Animals in High Dimensions

We present an implementation of Redelemeier's algorithm for the enumeration of lattice animals in high dimensional lattices. The implementation is lean and fast enough to allow us to extend the existing tables of animal counts, perimeter polynomials and series expansion coefficients in $d$-dimensional hypercubic lattices for $3 \leq d\leq 10$. From the data we compute formulas for perimeter polynomials for lattice animals of size $n\leq 11$ in arbitrary dimension $d$. When amended by combinatorial arguments, the new data suffices to yield explicit formulas for the number of lattice animals of size $n\leq 14$ and arbitrary $d$. We also use the enumeration data to compute numerical estimates for growth rates and exponents in high dimensions that agree very well with Monte Carlo simulations and recent predictions from field theory.Comment: 18 pages, 7 figures, 6 tables; journal versio

Critical Exponent for the Density of Percolating Flux

This paper is a study of some of the critical properties of a simple model for flux. The model is motivated by gauge theory and is equivalent to the Ising model in three dimensions. The phase with condensed flux is studied. This is the ordered phase of the Ising model and the high temperature, deconfined phase of the gauge theory. The flux picture will be used in this phase. Near the transition, the density is low enough so that flux variables remain useful. There is a finite density of finite flux clusters on both sides of the phase transition. In the deconfined phase, there is also an infinite, percolating network of flux with a density that vanishes as $T \rightarrow T_{c}^{+}$. On both sides of the critical point, the nonanalyticity in the total flux density is characterized by the exponent $(1-\alpha)$. The main result of this paper is a calculation of the critical exponent for the percolating network. The exponent for the density of the percolating cluster is $\zeta = (1-\alpha) - (\varphi-1)$. The specific heat exponent $\alpha$ and the crossover exponent $\varphi$ can be computed in the $\epsilon$-expansion. Since $\zeta < (1-\alpha)$, the variation in the separate densities is much more rapid than that of the total. Flux is moving from the infinite cluster to the finite clusters much more rapidly than the total density is decreasing.Comment: 20 pages, no figures, Latex/Revtex 3, UCD-93-2

Site percolation and random walks on d-dimensional Kagome lattices

The site percolation problem is studied on d-dimensional generalisations of the Kagome' lattice. These lattices are isotropic and have the same coordination number q as the hyper-cubic lattices in d dimensions, namely q=2d. The site percolation thresholds are calculated numerically for d= 3, 4, 5, and 6. The scaling of these thresholds as a function of dimension d, or alternatively q, is different than for hypercubic lattices: p_c ~ 2/q instead of p_c ~ 1/(q-1). The latter is the Bethe approximation, which is usually assumed to hold for all lattices in high dimensions. A series expansion is calculated, in order to understand the different behaviour of the Kagome' lattice. The return probability of a random walker on these lattices is also shown to scale as 2/q. For bond percolation on d-dimensional diamond lattices these results imply p_c ~ 1/(q-1).Comment: 11 pages, LaTeX, 8 figures (EPS format), submitted to J. Phys.

Hard squares with negative activity

We show that the hard-square lattice gas with activity z= -1 has a number of remarkable properties. We conjecture that all the eigenvalues of the transfer matrix are roots of unity. They fall into groups (``strings'') evenly spaced around the unit circle, which have interesting number-theoretic properties. For example, the partition function on an M by N lattice with periodic boundary condition is identically 1 when M and N are coprime. We provide evidence for these conjectures from analytical and numerical arguments.Comment: 8 page

Complexity of a complex trait locus: HP, HPR, haemoglobin and cholesterol

HP and HPR are related and contiguous genes in strong linkage disequilibrium (LD), encoding haptoglobin and haptoglobin-related protein. These bind and chaperone free Hb for recycling, protecting against oxidation. A copy number variation (CNV) within HP (Hp1/Hp2) results in different possible haptoglobin complexes which have differing properties. HPR rs2000999 (G/A), identified in meta-GWAS, influences total cholesterol (TC) and LDL-cholesterol (LDL-C). We examined the relationship between HP CNV, HPR rs2000999, Hb, red cell count (RCC), LDL-C and TC in the British Women's Heart and Health Study (n=2779 for samples having CNV, rs2000999, and phenotypes). Analysing single markers by linear regression, rs2000999 was associated with LDL-C (β=0.040 mmol/L, p=0.023), TC (β=-0.040 mmol/L, p=0.019), Hb (β=-0.044 g/dL, p=0.028) and borderline with RCC (β=-0.032 × 10(12)/L, p=0.066). Analysis of CNV by linear regression revealed an association with Hb (Hp1 vs Hp2, β=0.057 g/dL, p=0.004), RCC (β=0.045 × 10(12)/L, p=0.014), and showed a trend with LDL-C and TC. There were 3 principal haplotypes (Hp1-G 36%; Hp2-G 45%; Hp2-A 18%). Haplotype comparisons showed that LDL-C and TC associations were from rs2000999; Hb and RCC associations derived largely from the CNV. Distinct genotype-phenotype effects are evident at the genetic epidemiological level once LD has been analysed, perhaps reflecting HP-HPR functional biology and evolutionary history. The derived Hp2 allele of the HP gene has apparently been subject to malaria-driven positive selection. Haptoglobin-related protein binds Hb and apolipoprotein-L, i.e. linking HPR to the cholesterol system; and the HPR/apo-L complex is specifically trypanolytic. Our analysis illustrates the complex interplay between functions and haplotypes of adjacent genes, environmental context and natural selection, and offers insights into potential use of haptoglobin or haptoglobin-related protein as therapeutic agents.Philip A.I. Guthrie, Santiago Rodriguez, Tom R. Gaunt, Debbie A. Lawlor George Davey Smith, Ian N.M. Da

Trypanosoma cruzi IIc: phylogenetic and phylogeographic insights from sequence and microsatellite analysis and potential impact on emergent Chagas disease.

Trypanosoma cruzi, the etiological agent of Chagas disease, is highly genetically diverse. Numerous lines of evidence point to the existence of six stable genetic lineages or DTUs: TcI, TcIIa, TcIIb, TcIIc, TcIId, and TcIIe. Molecular dating suggests that T. cruzi is likely to have been an endemic infection of neotropical mammalian fauna for many millions of years. Here we have applied a panel of 49 polymorphic microsatellite markers developed from the online T. cruzi genome to document genetic diversity among 53 isolates belonging to TcIIc, a lineage so far recorded almost exclusively in silvatic transmission cycles but increasingly a potential source of human infection. These data are complemented by parallel analysis of sequence variation in a fragment of the glucose-6-phosphate isomerase gene. New isolates confirm that TcIIc is associated with terrestrial transmission cycles and armadillo reservoir hosts, and demonstrate that TcIIc is far more widespread than previously thought, with a distribution at least from Western Venezuela to the Argentine Chaco. We show that TcIIc is truly a discrete T. cruzi lineage, that it could have an ancient origin and that diversity occurs within the terrestrial niche independently of the host species. We also show that spatial structure among TcIIc isolates from its principal host, the armadillo Dasypus novemcinctus, is greater than that among TcI from Didelphis spp. opossums and link this observation to differences in ecology of their respective niches. Homozygosity in TcIIc populations and some linkage indices indicate the possibility of recombination but cannot yet be effectively discriminated from a high genome-wide frequency of gene conversion. Finally, we suggest that the derived TcIIc population genetic data have a vital role in determining the origin of the epidemiologically important hybrid lineages TcIId and TcIIe

Square lattice site percolation at increasing ranges of neighbor interactions

We report site percolation thresholds for square lattice with neighbor interactions at various increasing ranges. Using Monte Carlo techniques we found that nearest neighbors (N$^2$), next nearest neighbors (N$^3$), next next nearest neighbors (N$^4$) and fifth nearest neighbors (N$^6$) yield the same $p_c=0.592...$. At odds, fourth nearest neighbors (N$^5$) give $p_c=0.298...$. These results are given an explanation in terms of symmetry arguments. We then consider combinations of various ranges of interactions with (N$^2$+N$^3$), (N$^2$+N$^4$), (N$^2$+N$^3$+N$^4$) and (N$^2$+N$^5$). The calculated associated thresholds are respectively $p_c=0.407..., 0.337..., 0.288..., 0.234...$. The existing Galam--Mauger universal formula for percolation thresholds does not reproduce the data showing dimension and coordination number are not sufficient to build a universal law which extends to complex lattices.Comment: 4 pages, revtex

Bethe approximation for self-interacting lattice trees

In this paper we develop a Bethe approximation, based on the cluster variation method, which is apt to study lattice models of branched polymers. We show that the method is extremely accurate in cases where exact results are known as, for instance, in the enumeration of spanning trees. Moreover, the expressions we obtain for the asymptotic number of spanning trees and lattice trees on a graph coincide with analogous expressions derived through different approaches. We study the phase diagram of lattice trees with nearest-neighbour attraction and branching energies. We find a collapse transition at a tricritical theta point, which separates an expanded phase from a compact phase. We compare our results for the theta transition in two and three dimensions with available numerical estimates.Comment: 10 pages, 3 figures, to be published in Europhysics Letter