New evidence of earthquake precursory phenomena in the 17 January 1995 Kobe earthquake, Japan

Significant advances, both in the theoretical understanding of rupture processes in heterogeneous media and in the methodology for characterizing critical behavior, allows us to reanalyze the evidence for criticality and especially log-periodicity in the previously reported chemical anomalies that preceded the Kobe earthquake. The ion (Cl^-, K⁺, Mg⁺⁺, NO₃ ^- and SO₄ ^-) concentrations of ground-water issued from deep wells located near the epicenter of the 1995 Kobe earthquake are taken as proxies for the cumulative damage preceding the earthquake. Using both a parametric and non-parametric analysis, the five data sets are compared extensively to synthetic time series. The null-hypothesis that the patterns documented on these times series result from noise decorating a simple power law is rejected with a very high confidence level. Received 21 January 2000     

Nonmonotonic external field dependence of the magnetization in a finite Ising model: Theory and MC simulation

Using field theory and Monte Carlo (MC) simulation we investigate the finite-size effects of the magnetization M for the three-dimensional Ising model in a finite cubic geometry with periodic boundary conditions. The field theory with infinite cutoff gives a scaling form of the equation of state where is the reduced temperature, h is the external field and L is the size of system. Below and at the theory predicts a nonmonotonic dependence of f(x,y) with respect to at fixed and a crossover from nonmonotonic to monotonic behaviour when y is further increased. These results are confirmed by MC simulation. The scaling function f(x,y) obtained from the field theory is in good quantitative agreement with the finite-size MC data. Good agreement is also found for the bulk value at . Received 20 July 1999 and Received in final form 11 November 1999     

Series expansion study of quantum percolation on the square lattice

We study the site and bond quantum percolation model on the two-dimensional square lattice using series expansion in the low concentration limit. We calculate series for the averages of , where T _ij (E) is the transmission coefficient between sites i and j, for k=0, 1, , 5 and for several values of the energy E near the center of the band. In the bond case the series are of order p¹⁴ in the concentration p(some of those have been formerly available to order p¹⁰) and in the site case of order p¹⁶. The analysis, using the Dlog-Padé approximation and the techniques known as M1 and M2, shows clear evidence for a delocalization transition (from exponentially localized to extended or power-law-decaying states) at an energy-dependent threshold p _q(E) in the range , confirming previous results (e.g. and for bond and site percolation) but in contrast with the Anderson model. The divergence of the series for different kis characterized by a constant gap exponent, which is identified as the localization length exponent from a general scaling assumption. We obtain estimates of . These values violate the bound of Chayes et al. Received 28 February 2000     

A lattice gas coupled to two thermal reservoirs: Monte Carlo and field theoretic studies

We investigate the collective behavior of an Ising lattice gas, driven to non-equilibrium steady states by being coupled to two thermal baths. Monte Carlo methods are applied to a two-dimensional system in which one of the baths is fixed at infinite temperature. Both generic long range correlations in the disordered state and critical properties near the second order transition are measured. Anisotropic scaling, a key feature near criticality, is used to extract and some critical exponents. On the theoretical front, a continuum theory, in the spirit of Landau-Ginzburg, is presented. Being a renormalizable theory, its predictions can be computed by standard methods of -expansions and found to be consistent with simulation data. In particular, the critical behavior of this system belongs to a universality class which is quite different from the uniformly driven Ising model. Received 4 October 2000     

Violation of finite-size scaling in three dimensions

We reexamine the range of validity of finite-size scaling in the lattice model and the field theory below four dimensions. We show that general renormalization-group arguments based on the renormalizability of the theory do not rule out the possibility of a violation of finite-size scaling due to a finite lattice constant and a finite cutoff. For a confined geometry of linear size L with periodic boundary conditions we analyze the approach towards bulk critical behavior as at fixed for where is the bulk correlation length. We show that for this analysis ordinary renormalized perturbation theory is sufficient. On the basis of one-loop results and of exact results in the spherical limit we find that finite-size scaling is violated for both the lattice model and the field theory in the region . The non-scaling effects in the field theory and in the lattice model differ significantly from each other. Received 5 February 1999     

3d Monte Carlo simulation of site-bond continuum percolation of spheres

We present off-lattice Monte Carlo simulations of site-bond percolation of semi-penetrable spheres or, equivalently, of hard spheres with a finite bond range. We will show that the crucial parameter is the effective volume fraction ( φ_e), i.e. the volume that is occupied or within the bond range of at least one particle. For the equivalent system of semi-penetrable spheres 1 - φ_e is the porosity. The bond percolation threshold (p _b) can be described in terms of φ_e by a simple analytical expression: log(φ_e)/log(φ_ec) + log(p _b)/log(p _bc) = 1, with p _bc = 0.12 independent of the bond range and φ_ec a constant that decreases with increasing bond range. Received: 10 March 2003 / Accepted: 23 April 2003 / Published online: 21 May 2003 RID="a" ID="a"e-mail: jean-christophe.gimel@univ-lemans.fr     

A model for anomalous directed percolation

We introduce a model for the spreading of epidemics by long-range infections and investigate the critical behaviour at the spreading transition. The model generalizes directed bond percolation and is characterized by a probability distribution for long-range infections which decays in d spatial dimensions as . Extensive numerical simulations are performed in order to determine the density exponent and the correlation length exponents and for various values of . We observe that these exponents vary continuously with , in agreement with recent field-theoretic predictions. We also study a model for pairwise annihilation of particles with algebraically distributed long-range interactions. Received: 4 September 1998 / Accepted: 22 September 1998     

Hierarchy of critical exponents of currents on -simplex fractals

Using the symmetry of ( d +1)-simplex fractals with decimation number b =2, the current distribution has been determined. Then using the renormalization group technique, based on the independent Schur's invariant polynomials of current distributions, the multifractal spectrum of even moments of current distributions has been evaluated analytically up to order six for an arbitrary value of d. Also the scaling exponents of order 8 and order 10 have been calculated numerically up to d =30. Received: 19 November 1997 / Revised: 21 January 1998 / Accepted: 9 February 1998     

Casimir amplitudes in a quantum spherical model with long-range interaction

A d-dimensional quantum model system confined to a general hypercubical geometry with linear spatial size L and “temporal size” 1/T ( T - temperature of the system) is considered in the spherical approximation under periodic boundary conditions. For a film geometry in different space dimensions , where is a parameter controlling the decay of the long-range interaction, the free energy and the Casimir amplitudes are given. We have proven that, if , the Casimir amplitude of the model, characterizing the leading temperature corrections to its ground state, is . The last implies that the universal constant of the model remains the same for both short, as well as long-range interactions, if one takes the normalization factor for the Gaussian model to be such that . This is a generalization to the case of long-range interaction of the well-known result due to Sachdev. That constant differs from the corresponding one characterizing the leading finite-size corrections at zero temperature which for is . Received 3 June 1999 and Received in final form 16 August 1999     

Finite-size scaling in the theory above the upper critical dimension

We derive exact results for several thermodynamic quantities of the O ( n ) symmetric field theory in the limit in a finite d-dimensional hypercubic geometry with periodic boundary conditions. Corresponding results are derived for an O ( n ) symmetric model on a finite d-dimensional lattice with a finite-range interaction. The leading finite-size effects near T_c of the field-theoretic model are compared with those of the lattice model. For 2 < d < 4, the finite-size scaling functions are verified to be universal. For d > 4, significant lattice effects are found. Finite-size scaling in its usual simple form does not hold for d > 4 but remains valid in a generalized form with two reference lengths. The finite-size scaling functions of the field theory turn out to be nonuniversal whereas those of the lattice model are independent of the nonuniversal model parameters. In particular, the field-theoretic model exhibits finite-size effects whose leading exponents differ from those of the lattice model. The widely accepted lowest-mode approach is shown to fail for both the field-theoretic and the lattice model above four dimensions. Received: 20 October 1997 / Accepted: 5 March 1998     

Low temperature thermodynamics of inverse square spin models in one dimension

We present a field-theoretic renormalization group calculation in two loop order for classical O(N)-models with an inverse square interaction in the vicinity of their lower critical dimensionality one. The magnetic susceptibility at low temperatures is shown to diverge like with a=(N-2)/(N-1) and . From a comparison with the exactly solvable Haldane-Shastry model we find that the same temperature dependence applies also to ferromagnetic quantum spin chains. Received: 20 February 1998 / Revised: 27 April 1998 / Accepted: 30 April 1998     

A DMRG study of the -symmetric Heisenberg chain

The spin one-half Heisenberg chain with U q [ SU (2)] symmetry is studied via density-matrix renormalization. Ground-state energy and q-symmetric correlation functions are calculated for the non-Hermitian case with integer r. This gives bulk and surface exponents for (para)fermionic correlations in the related Ising and Potts models. The case of real q corresponding to a diffusion problem is treated analytically. Received: 18 February 1998 / Accepted: 17 March 1998     

An analytical method to compute an approximate value of the site percolation threshold P<Subscript>c</Subscript>

An analytical method to compute the site percolation threshold is introduced. This method yields an approximate value of larger or equal to the real value. As examples, the computation of is presented for 4 lattices in 2 dimensions: square, triangular, honeycomb and kagome. The results obtained are 0.592 871 6, 0.5, 0.765 069, 0.654 653 7, to be compared with the real values 0.592 746 0, 0.5, 0.697 043, 0.652 703 6. The method is not limited to 2 dimensions. Received 27 July 1999 and Received in final form 29 November 1999     

Excitations in the dilute lattice model: , and mass spectra

On the basis of features observed in the exact perturbation approach solution for the eigenspectrum of the dilute A₃ model, we propose expressions for excitations in the dilute A₄ and A₆ models. Principally, we require that these expressions satisfy the appropriate inversion relations. We demonstrate that they give the expected E₇ and E₆ mass spectra, and universal amplitudes, and agree with numerical expressions for the eigenvalues. Received: 17 February 1998 / Accepted: 30 April 1998     

Finite-size scaling in systems with long-range interaction

The finite-size critical properties of the (n) vector ϕ⁴ model, with long-range interaction decaying algebraically with the interparticle distance r like r ^{-d - σ}, are investigated. The system is confined to a finite geometry subject to periodic boundary condition. Special attention is paid to the finite-size correction to the bulk susceptibility above the critical temperature T _c. We show that this correction has a power-law nature in the case of pure long-range interaction i.e. 0 < σ < 2 and it turns out to be exponential in case of short-range interaction i.e.σ = 2. The results are valid for arbitrary dimension d, between the lower ( d _< = σ) and the upper ( d _> = 2σ) critical dimensions. Received 2 July 2001 and Received in final form 4 Septembre 2001     

Charge orderings in the atomic limit of the extended Hubbard model

The extended Hubbard model in the atomic limit (AL-EHM) on a square lattice with periodic boundary conditions is studied with use of the Monte Carlo (MC) method. Within the grand canonical ensemble the phase and order-order boundaries for charge orderings are obtained. The phase diagrams include three types of charge ordered phases and the nonordered phase. The system exhibits very rich structure and shows unusual multicritical behavior. In the limiting case of t_ij=0, the EHM is equivalent to the pseudospin model with single-ion anisotropy , exchange interaction W in an effective magnetic field . This classical spin model is analyzed using the MC method for the canonical ensemble. The phase diagram is compared with the known results for the Blume-Capel model.     

Universality in three dimensional random-field ground states

We investigate the critical behavior of three-dimensional random-field Ising systems with both Gauss and bimodal distribution of random fields and additional the three-dimensional diluted Ising antiferromagnet in an external field. These models are expected to be in the same universality class. We use exact ground-state calculations with an integer optimization algorithm and by a finite-size scaling analysis we calculate the critical exponents , , and . While the random-field model with Gauss distribution of random fields and the diluted antiferromagnet appear to be in same universality class, the critical exponents of the random-field model with bimodal distribution of random fields seem to be significantly different. Received: 9 July 1998 / Received in final form: 15 July 1998 / Accepted: 20 July 1998     

The kinetic spin-1 Blume-Capel model with competing dynamics

We have studied by means of Monte-Carlo simulation and exact finite-size analysis, the spin-1 Blume Capel model with Glauber and Kawasaki dynamics. The Kawasaki spin-exchange process transfers energy into the system from an external source. Some phase diagrams of the model are presented. For some parameter values, the system displays a kind of self-organization phenomenon within the disordered phase. Received 15 February 2000