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

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

Double Parton Scattering Singularity in One-Loop Integrals

We present a detailed study of the double parton scattering (DPS) singularity, which is a specific type of Landau singularity that can occur in certain one-loop graphs in theories with massless particles. A simple formula for the DPS singular part of a four-point diagram with arbitrary internal/external particles is derived in terms of the transverse momentum integral of a product of light cone wavefunctions with tree-level matrix elements. This is used to reproduce and explain some results for DPS singularities in box integrals that have been obtained using traditional loop integration techniques. The formula can be straightforwardly generalised to calculate the DPS singularity in loops with an arbitrary number of external particles. We use the generalised version to explain why the specific MHV and NMHV six-photon amplitudes often studied by the NLO multileg community are not divergent at the DPS singular point, and point out that whilst all NMHV amplitudes are always finite, certain MHV amplitudes do contain a DPS divergence. It is shown that our framework for calculating DPS divergences in loop diagrams is entirely consistent with the `two-parton GPD' framework of Diehl and Schafer for calculating proton-proton DPS cross sections, but is inconsistent with the `double PDF' framework of Snigirev.Comment: 29 pages, 8 figures. Minor corrections and clarifications added. Version accepted for publication in JHE

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

Intracortical microstimulation of human somatosensory cortex induces natural perceptual biases

Time-order error, a psychophysical phenomenon in which the duration in between successive stimuli alters perception, has been studied for decades by neuroscientists and psychologists. To date, however, the locus of these effects is unknown. We use intracortical microstimulation of somatosensory cortex in three humans with spinal cord injury as a tool to bypass initial stages of processing and restrict the possible locations that signals could be modified. Using a 2-interval forced choice amplitude discrimination paradigm, we first assessed the extent to which order effects are observed. Comparing trials where the standard stimulus was in the first or second interval, we found that systematic biases are exhibited, typically causing the intensity of the second stimulus to be overestimated The degree of this overestimation for individual electrodes was dependent on the perceptual sensitivity to changes in stimulus amplitude. To investigate the role of memory on this phenomenon, we implemented a 2-interval magnitude estimation task in which participants were instructed to ignore the first stimulus and again found that the perceptual intensity of the second stimulus tended to be enhanced by the first in a manner that depended on the amplitude and duration of the first stimulus. Finally, we repeated both paradigms while varying the inter-stimulus interval to examine the timescale over which these effects occur and found that longer inter-stimulus intervals reduced the effect size. These results show that direct activation of primary somatosensory cortex is sufficient to induce time-order errors

Relaxation properties in a lattice gas model with asymmetrical particles

We study the relaxation process in a two-dimensional lattice gas model, where the interactions come from the excluded volume. In this model particles have three arms with an asymmetrical shape, which results in geometrical frustration that inhibits full packing. A dynamical crossover is found at the arm percolation of the particles, from a dynamical behavior characterized by a single step relaxation above the transition, to a two-step decay below it. Relaxation functions of the self-part of density fluctuations are well fitted by a stretched exponential form, with a $\beta$ exponent decreasing when the temperature is lowered until the percolation transition is reached, and constant below it. The structural arrest of the model seems to happen only at the maximum density of the model, where both the inverse diffusivity and the relaxation time of density fluctuations diverge with a power law. The dynamical non linear susceptibility, defined as the fluctuations of the self-overlap autocorrelation, exhibits a peak at some characteristic time, which seems to diverge at the maximum density as well.Comment: 7 pages and 9 figure

The Oscillatory Behavior of the High-Temperature Expansion of Dyson's Hierarchical Model: A Renormalization Group Analysis

We calculate 800 coefficients of the high-temperature expansion of the magnetic susceptibility of Dyson's hierarchical model with a Landau-Ginzburg measure. Log-periodic corrections to the scaling laws appear as in the case of a Ising measure. The period of oscillation appears to be a universal quantity given in good approximation by the logarithm of the largest eigenvalue of the linearized RG transformation, in agreement with a possibility suggested by K. Wilson and developed by Niemeijer and van Leeuwen. We estimate $\gamma$ to be 1.300 (with a systematic error of the order of 0.002) in good agreement with the results obtained with other methods such as the $\epsilon$-expansion. We briefly discuss the relationship between the oscillations and the zeros of the partition function near the critical point in the complex temperature plane.Comment: 21 pages, 10 Postcript figures, latex file, uses revte

First- and second-order phase transitions in a driven lattice gas with nearest-neighbor exclusion

A lattice gas with infinite repulsion between particles separated by $\leq 1$ lattice spacing, and nearest-neighbor hopping dynamics, is subject to a drive favoring movement along one axis of the square lattice. The equilibrium (zero drive) transition to a phase with sublattice ordering, known to be continuous, shifts to lower density, and becomes discontinuous for large bias. In the ordered nonequilibrium steady state, both the particle and order-parameter densities are nonuniform, with a large fraction of the particles occupying a jammed strip oriented along the drive. The relaxation exhibits features reminiscent of models of granular and glassy materials.Comment: 8 pages, 5 figures; results due to bad random number generator corrected; significantly revised conclusion

Series studies of the Potts model. I: The simple cubic Ising model

The finite lattice method of series expansion is generalised to the $q$-state Potts model on the simple cubic lattice. It is found that the computational effort grows exponentially with the square of the number of series terms obtained, unlike two-dimensional lattices where the computational requirements grow exponentially with the number of terms. For the Ising ($q=2$) case we have extended low-temperature series for the partition functions, magnetisation and zero-field susceptibility to $u^{26}$ from $u^{20}$. The high-temperature series for the zero-field partition function is extended from $v^{18}$ to $v^{22}$. Subsequent analysis gives critical exponents in agreement with those from field theory.Comment: submitted to J. Phys. A: Math. Gen. Uses preprint.sty: included. 24 page

Multi-Parton Interactions at the LHC

We review the recent progress in the theoretical description and experimental observation of multiple parton interactions. Subjects covered include experimental measurements of minimum bias interactions and of the underlying event, models of soft physics implemented in Monte Carlo generators, developments in the theoretical description of multiple parton interactions and phenomenological studies of double parton scattering. This article stems from contributions presented at the Helmholtz Alliance workshop on "Multi-Parton Interactions at the LHC", DESY Hamburg, 13-15 September 2010.Comment: 68 page