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

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.

Ising spins coupled to a four-dimensional discrete Regge skeleton

Regge calculus is a powerful method to approximate a continuous manifold by a simplicial lattice, keeping the connectivities of the underlying lattice fixed and taking the edge lengths as degrees of freedom. The discrete Regge model employed in this work limits the choice of the link lengths to a finite number. To get more precise insight into the behavior of the four-dimensional discrete Regge model, we coupled spins to the fluctuating manifolds. We examined the phase transition of the spin system and the associated critical exponents. The results are obtained from finite-size scaling analyses of Monte Carlo simulations. We find consistency with the mean-field theory of the Ising model on a static four-dimensional lattice.Comment: 19 pages, 7 figure

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

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

Specific heat amplitude ratios for anisotropic Lifshitz critical behaviors

We determine the specific heat amplitude ratio near a $m$-axial Lifshitz point and show its universal character. Using a recent renormalization group picture along with new field-theoretical $\epsilon_{L}$-expansion techniques, we established this amplitude ratio at one-loop order. We estimate the numerical value of this amplitude ratio for $m=1$ and $d=3$. The result is in very good agreement with its experimental measurement on the magnetic material $MnP$. It is shown that in the limit $m \to 0$ it trivially reduces to the Ising-like amplitude ratio.Comment: 8 pages, RevTex, accepted as a Brief Report in Physical Review

Series Expansion Calculation of Persistence Exponents

We consider an arbitrary Gaussian Stationary Process X(T) with known correlator C(T), sampled at discrete times T_n = n \Delta T. The probability that (n+1) consecutive values of X have the same sign decays as P_n \sim \exp(-\theta_D T_n). We calculate the discrete persistence exponent \theta_D as a series expansion in the correlator C(\Delta T) up to 14th order, and extrapolate to \Delta T = 0 using constrained Pad\'e approximants to obtain the continuum persistence exponent \theta. For the diffusion equation our results are in exceptionally good agreement with recent numerical estimates.Comment: 5 pages; 5 page appendix containing series coefficient

Causality and defect formation in the dynamics of an engineered quantum phase transition in a coupled binary Bose-Einstein condensate

Continuous phase transitions occur in a wide range of physical systems, and provide a context for the study of non-equilibrium dynamics and the formation of topological defects. The Kibble-Zurek (KZ) mechanism predicts the scaling of the resulting density of defects as a function of the quench rate through a critical point, and this can provide an estimate of the critical exponents of a phase transition. In this work we extend our previous study of the miscible-immiscible phase transition of a binary Bose-Einstein condensate (BEC) composed of two hyperfine states in which the spin dynamics are confined to one dimension [J. Sabbatini et al., Phys. Rev. Lett. 107, 230402 (2011)]. The transition is engineered by controlling a Hamiltonian quench of the coupling amplitude of the two hyperfine states, and results in the formation of a random pattern of spatial domains. Using the numerical truncated Wigner phase space method, we show that in a ring BEC the number of domains formed in the phase transitions scales as predicted by the KZ theory. We also consider the same experiment performed with a harmonically trapped BEC, and investigate how the density inhomogeneity modifies the dynamics of the phase transition and the KZ scaling law for the number of domains. We then make use of the symmetry between inhomogeneous phase transitions in anisotropic systems, and an inhomogeneous quench in a homogeneous system, to engineer coupling quenches that allow us to quantify several aspects of inhomogeneous phase transitions. In particular, we quantify the effect of causality in the propagation of the phase transition front on the resulting formation of domain walls, and find indications that the density of defects is determined during the impulse to adiabatic transition after the crossing of the critical point.Comment: 23 pages, 10 figures. Minor corrections, typos, additional referenc

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

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