Low-density Series Expansions Research Articles

Low-density series expansions for directed percolation: IV. Temporal disorder

We introduce a model for temporally disordered directed percolation in which the probability of spreading from a vertex $(t,x)$, where $t$ is the time and $x$ is the spatial coordinate, is independent of $x$ but depends on $t$. Using a very efficient algorithm we calculate low-density series for bond percolation on the directed square lattice. Analysis of the series yields estimates for the critical point $p_c$ and various critical exponents which are consistent with a continuous change of the critical parameters as the strength of the disorder is increased.

Low-density series expansions for directed percolation: III. Some two-dimensional lattices

We use very efficient algorithms to calculate low-density series for bond and site percolation on the directed triangular, honeycomb, kagom\'e, and $(4.8^2)$ lattices. Analysis of the series yields accurate estimates of the critical point $p_c$ and various critical exponents. The exponent estimates differ only in the $5^{th}$ digit, thus providing strong numerical evidence for the expected universality of the critical exponents for directed percolation problems. In addition we also study the non-physical singularities of the series.

Some Properties of Weight Factors arising in Low-Density Series Expansion for Percolation Models

Let F ( G ) be any additive property of a simple graph such that F ( G )= F ( G 1 )+ F ( G 2 ), where G is the series combination of graphs G 1 and G 2 . The weight factor W ( G ) which is based on F ( G ) arises in the low-density series expansion techniques for percolation models as \(W(G)=\sum_{G^{\prime}\subseteq G}(-1)^{e-e^{\prime}}F(G^{\prime})\eta(G^{\prime})\), where η( G ′ ) is the indicator that G ′ cover-able sub-graph or without dangling ends. The purpose of this paper is to prove the weight factor formula for additive property of F as W ( G )= d ( G 2 ) W ( G 1 )+ d ( G 1 ) W ( G 2 ), where d ( G 1 ) are d ( G 2 ) the d-weight for graphs G 1 and G 2 respectively. This result will be more simplified in the case of Directed Percolation Models using Mobius function property. A new few formulas for the resistive weight factors are also derived for a graph, which is parallel combination of n edges.

Low-Density Series Expansion for the Domany-Kinzel Model

Domany-Kinzel (DK) model is a family of the 1+1 dimensional stochastic cellular automata with two parameters p 1 and p 2 , which simulate time evolution of interacting active elements in a random medium. By identifying a set of active sites on the spatio-temporal plane with a percolation cluster, we discuss the directed percolation (DP) transitions in the DK model. We parameterize p 1 = p and p 2 = α p with p ∈[0,1] and α∈[0,2] and calculate the mean cluster size and other quantities characterizing the DP cluster as the series of p up to order 51 for several values of α by using a graphical expansion formula recently given by Konno and Katori. We analyze the series by the first- and second-order differential approximations and the Zinn-Justin method and study the dependence on α of the convergence of estimations of critical values and critical exponents. In the mixed site-bond DP region, 1 ≤α≤ 1.3553, the convergence is excellent. As α→2 slowing down of convergence and as α→0 peculiar oscillation of estim...

Low-density series expansions for directed percolation: II. The square lattice with a wall

A new algorithm for the derivation of low-density expansions has been used to greatly extend the series for moments of the pair-connectedness on the directed square lattice near an impenetrable wall. Analysis of the series yields very accurate estimates for the critical point and exponents. In particular, the estimate for the exponent characterizing the average cluster length near the wall, $\tau_1=1.00014(2)$, appears to exclude the conjecture $\tau_1=1$. The critical point and the exponents $\nu_{\parallel}$ and $\nu_{\perp}$ have the same values as for the bulk problem.

Low-density series expansions for directed percolation: I. A new efficient algorithm with applications to the square lattice

A new algorithm for the derivation of low-density series for percolation on directed lattices is introduced and applied to the square lattice bond and site problems. Numerical evidence shows that the computational complexity grows exponentially, but with a growth factor $\lambda < \protect{\sqrt[8]{2}}$, which is much smaller than the growth factor $\lambda = \protect{\sqrt[4]{2}}$ of the previous best algorithm. For bond (site) percolation on the directed square lattice the series has been extended to order 171 (158). Analysis of the series yields sharper estimates of the critical points and exponents.

Series expansion analysis of the backbone properties of two-dimensional percolation clusters

Low-density series expansions for the backbone properties of two-dimensional bond percolation clusters are derived and analysed. Expansions for most of the 14 properties considered are new and are obtained to order on the square lattice and order on the triangular lattice. Earlier series work was confined to three properties of the square lattice and was to order . The fractal dimension of the bonds or sites in the backbone is estimated to be and is intermediate between a previously conjectured field theory value and the latest Monte Carlo results. The union, intersection and length of the longest self-avoiding paths are found to have the same fractal dimension which is close to and consistent with the field theory conjecture for . On the other hand, the union intersection and length of the shortest paths are found to have different dimensions and in the case of the intersection, the triangular and square lattices are found to have sigificantly different dimensions. The fractal dimension of the shortest path also appears to be non-universal and we find for the square lattice and for the triangular lattice. Critical amplitude ratios are considered and found to be in agreement with theoretical inequalities.

London equation of state for quantum hard disks

For a two-dimensional hard-disk boson system we propose an analytical London-type expression for the ground-state energy as a function of density ranging from zero to close packing. Such an equation interpolates between the leading term of the exact low-density series expansion and the behavior expected at close-packing based on uncertainty-principle arguments and determined by the correct value of the residue which is a dimensionless constant whose value depends on the space dimension as well as on the kind of close packing configuration involved. As for any reasonably accurate equation of state, variational Monte Carlo calculations for both fluid and crystalline energy branches turn out to be close upper bounds to the resulting so-called London–Hubbard energy which includes pair-correlation effects.

Low-density series expansions for directed percolation on square and triangular lattices

Greatly extended series have been derived for moments of the pair-connectedness for bond and site percolation on the directed square and triangular lattices. The length of the various series has been at least doubled to more than 110 (100) terms for the square-lattice bond (site) problem and more than 55 terms for the bond and site problems on the triangular lattice. Analysis of the series leads to very accurate estimates for the critical parameters and generally seems to rule out simple rational values for the critical exponents. The values of the critical exponents for the average cluster size, parallel and perpendicular connectedness lengths are estimated by D 2:277 69.4/, k D 1:733 825.25/ and ? D 1:096 844.14/, respectively. An improved estimate for the percolation probability exponent is obtained from the scaling relation D.k C ? /= 2 D 0 :276 49.4/. In all cases the leading correction to scaling term is analytic.

Quantum-Hard-Sphere System Equations of State Revisited

Analytical equations of state for boson and fermion hard-sphere fluids ranging from very low to very high densities are constructed. Such equations of state serve as a zero-order (reference) state upon which to build so-called quantum-thermodynamic-perturbation corrections in describing real but simple quantum fluids at zero temperature. The fluid branch extrapolations from the exact low-density series expansions for the energy are carried out by incorporating various physical arguments, such as close packing densities and residues. Modified London equations of state for the high-density crystalline branch agree very well with computer simulations, and at close packing with certain experimental results at high pressure.

New series expansion data for surface and bulk resistivity and conductivity in two-dimensional percolation

Low-density series expansions are obtained for random resistor networks. The expansions are obtained on the square lattice to order 18 and on the triangular lattice to order 14. Bulk and surface expansions are given for both resistive and conductive susceptibilities. The balance of the evidence obtained from analysing these series is in favour of the value for the critical exponent of the resistance scale and supports the existence of only a single such scale for bulk and surface susceptibilities. This value is in agreement with earlier Monte Carlo work.

Counting lattice animals: A parallel attack

A parallel algorithm for the enumeration of isolated connected clusters on a regular lattice is presented. The algorithm has been implemented on 17 RISC-based workstations to calculate the perimeter polynomials for the plane triangular lattice up to clustersizes=21. New data for perimeter polynomials Ds up toD 21, total number of clustersg s up tog 22, and coefficientsb r in the low-density series expansion of the mean cluster size up tob 21 are given.

Series expansion study of the distribution of currents in the elements of a random diode-insulator network

The critical exponent zeta k of the kth moment of the current distribution for random diode insulator networks on the square and simple cubic lattices is calculated for a range of k using low density series expansion techniques. It is also shown that zeta 1= nu /sub /// where nu /sub /// is the critical exponent of the parallel connectedness length for directed percolation The values of zeta k for the simple cubic lattice are well fitted by a simple exponential formula with ( zeta k-1)/( zeta k+1-1)=1/2. The Skal=Shklovskii scaling relation for the conductivity is generalised to the kth moment and it follows that the exponent K describing the divergence of flicker noise is given by kappa approximately=(d-1) nu perpendicular to + nu /sub ///+ zeta 4-2 zeta 2. The authors estimate this for both lattices.

Multiple exchange inHe3and in the Wigner solid

We use a multidimensional WKB calculation to determine the leading terms of a high-density series expansion for the multiple-exchange frequencies in solid $^{3}\mathrm{He}$. We also calculate, within the same formulation, the first terms of a low-density series expansion for exchange in the two-dimensional Wigner solid of electrons. The hierarchy between the exchange frequencies is dominated by the geometry of the lattice. Triple exchange is preponderant in the two-dimensional triangular lattice. This result holds for potentials as different as the ${(\frac{\ensuremath{\sigma}}{r})}^{12}$ (solid $^{3}\mathrm{He}$) and the Coulomb potential (Wigner solid). Three-particle exchange also dominates in the hcp lattice. We find for a hypothetical high-density bcc $^{3}\mathrm{He}$ solid the same hierarchy as that deduced from the experimental results at physical densities: The planar four-particle exchange ${K}_{P}$ and the triple exchange ${J}_{t}$ dominate, the folded four-particle exchange ${K}_{F}$ is much lower. For both ${K}_{P}$ and ${J}_{t}$ the lengths of the exchange paths and the tunneling barrier heights are practically the same. We expect these results to be qualitatively valid at physical densities and understand why ${K}_{P}$ and ${J}_{t}$ depend similarly on the molar volume.

A note on Coniglio's lemma

It is proved that Coniglio's (1981, 1982) lemma holds for any pair of points in any graph in the low-density series expansion of the cut-edge weighted pair connectedness.

Calculation of the Green's function from high- and low-density series expansions for disordered transport

We investigate density expansions for the configurationally averaged Green's function for a random walk on a (site) disordered lattice. Two-point Pad\'e summation techniques are used in conjunction with scaling arguments to examine behavior near the percolation density. Recent proposals for the structure of the percolation cluster are discussed in light of the results.

Directed percolation: series expansion for some three-dimensional lattices

Low density series expansions have been obtained for the directed cubic lattice bond problem and the directed cubic and body centred cubic lattice site problems. Estimates of the critical probabilities and critical exponents gamma , gamma 0, nu //, nu perpendicular to for these problems are obtained. The exponent values are found to be consistent with the universality of directed percolation and Reggeon field theory and the hyperscaling relation beta =(D nu perpendicular to + nu //- gamma )/2.

Directed percolation: mean field theory and series expansions for some two-dimensional lattices

The shape of percolation clusters in the directed percolation problem is studied within mean field theory. Low-density series expansions are obtained for bond and site problems on the square and triangular lattices. These are used to test a scaling formula for the pair connectedness which involves two lengths xi /sub ///( rho ) and xi perpendicular to ( rho ). Determination of the exponents nu /sub /// and nu perpendicular to for these lengths shows that all four problems are in the same universality class and the values support the hyperscaling relation beta =1/2(D nu perpendicular to + nu /sub ///- gamma ). An independent argument is given for this relation.

Series analysis study of bond-site percolation on an anisotropic triangular lattice

We derive low-density series expansions for the mean size of finite clusters on an anisotropic triangular lattice. By varying a bond density parameter this model includes site percolation on the square and triangular lattices and interpolates smoothly between the two. We identify the critical line and our results are consistent with the critical exponent y being constant along this line.

Series expansion studies of percolation at a surface

Estimates of the local exponents gamma 1 and gamma 11 for the ordinary transition in the bond problem on the triangular lattice and bond and site problems on the face-centred cubic lattice, are obtained from low-density series expansions. The authors' data are consistent with the scaling prediction gamma + nu =2 gamma 1+ gamma 11 and the assumption that the ordinary transition correlation length diverges with its homogeneous exponent. Their results indicate that a surface transition occurs on the FCC lattice but not on the triangular lattice. Analogous results for the self-avoiding walk problem are also discussed.