Magnetization of the Ising model on the generalized checkerboard lattice

We consider the Ising model on the generalized checkerboard lattice. Using a recent result by Baxter and Choy, we derive exact expressions for the magnetization of nodal spins at two values of the magnetic field,H=0 andH=i1/2kT. Our results are given in terms of Boltzmann weights of a unit cell of the checkerboard lattice without specifying its cell structures.     

LARGE-SCALE PATHOLOGY OF UNIVERSES WITH VII0×VIII ISOMETRIES

The goal of this paper is to show somewhat unexpected globally pathologic properties in universes described by a class of static planary symmetric exact solutions with G ₆-group of motion. In order to achieve this aim, the Killing vectors, the null geodesics and the Penrose diagrams corresponding to different expressions of g ₄₄=–e ^2f(z), with f(z) solutions of Einstein's equations, have been employed. Finally, woking in a particular gauge, we focus on the behaviour of the radiating electromagnetic modes and derive the observable components of E and B and the expressions of the essential component of the Umov-Poynting vector.     

The spherical limit forn-vector correlations

We investigate then limit of then-vector model single-spin and pairspin correlation functions. In this limit we show that the correlation functions become those of the corresponding spherical model.     

Simple and multiple scattering in disordered one-dimensional media: Renormalisation group approach

The exact characteristic penetration length associated with both simple and multiple incoherent elastic scattering in semi-infinite one-dimensional disordered media is established as a function ofp (concentration of scattering centers) andf ₀ (transmission coefficient of a single center). Then we exhibit how these phenomena can be seen as critical ones, and the corresponding are reobtained within convenient real space renormalisation group frameworks. Finally we discuss a generalized model where the single center transmission coefficientf can randomly take two different valuesf ₁ andf ₂.     

H-theorem for the (modified) nonlinear Enskog equation

We construct anH-function suitable for a system of dense hard spheres satisfying the (modified) nonlinear Enskog equation and we show that _t H 0. The equality sign holds only when the system has reached absolute equilibrium, in which caseS=– k_B H becomes the exact equilibrium entropy of the hard-sphere fluid.     

Migdal-Kadanoff renormalization group for theO(n) model

We perform a Migdal-Kadanoff renormalization group calculation on anO(n) symmetric model on ad-dimensional hypercubic lattice,d=2, 3. We find that in two dimensions the critical fixed point disappears asn=n _KT1.96, which is in good agreement with the exact valuen _KT=2. In three dimensions the fixed point persists much longer, albeit not all the way up to infinity. Surface critical phenomena in a semiinfiniteO(n) model are also considered.     

Exact Mean-Field Solutions of the Asymmetric Random Average Process

We consider the asymmetric random average process (ARAP) with continuous mass variables and parallel discrete time dynamics studied recently by Krug/Garcia and Rajesh/Majumdar [both J. Statist. Phys. 99 (2000)]. The model is defined by an arbitrary state-independent fraction density function (r) with support on the unit interval. We examine the exactness of mean-field steady-state mass distributions in dependence of and identify as a conjecture based on high order calculations the class of density functions yielding product measure solutions. Additionally the exact form of the associated mass distributions P(m) is derived. Using these results we show examplary the exactness of the mean-field ansatz for monomial fraction densities (r)=(n–1) r ^n–2 with n2. For verification we calculate the mass distributions P(m) explicitly and prove directly that product measure holds. Furthermore we show that even in cases where the steady state is not given by a product measure very accurate approximants can be found in the class .     

Error bound for the Hartree-Fock energy of atoms and molecules

We estimate the error of the Hartree-Fock energy of atoms and molecules in terms of the one-particle density matrix corresponding to the exact ground state. As an application we show this error to be of orderO(Z ^5/3–) for any <2/21 as the total nuclear chargeZ becomes large.     

On the Large-Distance Behavior of Correlations for a Hierarchical N-Component Classical Vector Model in Three Dimensions

We consider the correlation functions for a hierarchical N-component classical vector model in three dimensions. For N = , we find explicitly the eigenvalues and global eigenfunctions of the linearized renormalization group transformation. In a very direct way, this yields the correlation functions for the N = model. In particular, we check that the two-point function has canonical decay.     

Anomalous fluctuations in random walk dynamics

The anomalous dispersion of noninteracting particles randomly walking in a network is considered. It is shown that the existence of large dangling branches attached to a backbone induces a l/f-like behavior in the current autocorrelation function at low frequencies. The waiting times associated with dangling loops scale liket ^–3/2. The size of the dangling branches provides a lower cutoff to the power law behavior. When the side branches are infinite, self-similar structures, the power law behavior persists up to a zero frequency. The currents we consider are created either by a bias on the random walk or by a current source. We consider both the total current, which is often referred to in the literature, and the current measured at endpoints of a specimen attached to a (model) battery. The differences and similarities between the two corresponding correlations are analyzed. In particular, we find that in the second case l/f noise exists only for large bias. When a statistical distribution of dangling branches is considered, we find that the largest power of frequency in the spectrum is 1.13. Much of our results are true when the dangling branches are replaced by traps having waiting time distributions that equal those of the branches. The waiting time associated with a power law distribution of dangling loops (m ^–x:m is the length of the loop) scales liket ^–1 –(x/2). However, it is shown that geometry alone can be responsible for the appearance of power laws in the spectra. Random geometry can be regarded as a model (or source) of random hopping times.     

Integral Presentations for the Universal R-Matrix

We present an integral formula for the universal R-matrix of quantum affine algebra U _q with Drinfeld comultiplication. We show that the properties of the universal R-matrix follow from the factorization properties of the cycles in proper configuration spaces. For general g we conjecture that such cycles exist and unique. For U _q we describe precisely the cycles and present a new simple expression for the universal R-matrix as a result of calculation of corresponding integrals.     

Exact solutions for the discrete Boltzmann models with specular reflection

We construct exact (1+1)-dimensional solutions (space x, time t), in the presence of a purely reflecting well, for both the four velocity discrete Boltzmann model and the Broadwell model. These exact solutions, sums of two similarity shock waves, are positive for x0, t0.     

Abelian model of gravitating magnetic monopole

A self-consistentU(1)-gauge model in gravitational field is investigated. The exact solutions of two types of corresponding field equations are obtained. These solutions can be interpreted as magnetic monopoles. The first solution is regular forr 0 and provides an everywhere regular geometry, the second one has a physical singularity. In order to guarantee the stability of the monopoles we introduce the notion of a gravitational topological charge using de Rham's cohomology theory. This topological charge describes the sizes and the inner structure of the monopole.     

ℤ n Baxter model: Symmetries and the Belavin parametrization

The _n Baxter model is an exactly solvable lattice model in the special case of the Belavin parametrization. For this parametrization we calculate the partition function,, in an antiferromagnetic region and the order parameter in a ferromagnetic region. We find that the order parameter is expressible in terms of a modular function of leveln which forn = 2 is the Onsager-Yang-Baxter result. In addition we determine the symmetry group of the finite lattice partition function for the general _n Baxter model.     

Correlation inequalities for vector spin models

Correlation inequalities forn-vector spin models (n 2) are reviewed. A relatively simple and unified derivation of the inequalities is achieved, using duplicate variable methods, for spin dimensionalitiesn=2 (plane rotator model),n=3 (classical Heisenberg model), andn=4. Although correlation inequalities are lacking forn > 4, new proofs are presented for the comparison inequalities relating correlations for systems with arbitrary spin dimensionality to corresponding correlations for systems with low spin dimensionality (n = 1 or 2).Research supported by National Science Foundation under Grant DMR 76-23071.     

Wetting in Potts and Blume-Capel models

We discuss the wetting of the interface between two ordered phases by the disordered one in the Potts model withq large. We argue that a low-temperature expansion can be used in this situation, with logq replacing. This model is analogous to the Blume-Capel model at low temperatures, which we use as an example to review the low-temperature expansions.     

Scale-Invariant Branch Distribution from a Soluble Stochastic Model

We consider a general model of branch competition that automatically leads to a critical branching configuration. This model is inspired by the 4– expansion of the dielectric breakdown model, but the mechanism of arriving at the critical point may be of relevance to other branching systems as well, such as fractures. The exact solution of this model clarifies the direct renormalization procedure used for the dielectric breakdown model, and demonstrates nonperturbatively the existence of additional irrelevant operators with complex scaling dimensions leading to discrete scale invariance. The anomalous exponents are shown to depend upon the details of branch interaction; we contrast with the branched growth model in which these exponents are universal to lowest order in 1–, and show that the branched growth model includes an inherent branch interaction different from that found in the dielectric breakdown model. We consider stationary and non-stationary regimes, corresponding to different growth geometries in the dielectric-breakdown model.     

Riccati-coupled similarity shock wave solutions for multispeed discrete Boltzmann models

We study nonstandard shock wave similarity solutions for three multispeed discrete Boltzmann models: (1) the square 8_i, model with speeds 1 and 2 with thex axis along one median, (2) the Cabannes cubic 14_i model with speeds 1 and 3 and thex axis perpendicular to one face, and (3) another 14_i, model with speeds 1 and 2. These models have five independent densities and two nonlinear Riccati-coupled equations. The standard similarity shock waves, solutions of scalar Riccati equations, are monotonic and the same behavior holds for the conservative macroscopic quantities. First, we determine exact similarity shock-wave solutions of coupled Riccati equations and we observe nonmonotonic behavior for one density and a smaller effect for one conservative macroscopic quantity when we allow a violation of the microreversibility. Second, we obtain new results on the Whitham weak shock wave propagation. Third, we solve numerically the corresponding dynamical system, with microreversibility satisfied or not, and we also observe the analogous nonmonotonic behavior.     

The 1/D expansion for low-dimensional classical magnets

The physical characteristics of two-dimensional classical ferro- and antiferro-magnets have been calculated in the whole temperature range by an analytical approach based on the expansion in powers of 1/D, whereD is the number of spin components. This approach works rather well since it yields exact results for antiferromagnetic susceptibility and specific heat atT=0 already in the first order in 1/D and it is consistent with HTSE at high temperatures. For the quantities singular atT=0, such as ferromagnetic susceptibility and correlation length, the 1/D expansion supports their general-D functional form in the low-temperature range obtained by Fukugita and Oyanagi. The critical index calculated in the first order in 1/D proves to be temperature dependent: =20/(D) (=T/T _c ^(MFT) ,T _c ^(MFT) =J ₀/D, J ₀ is the zero Fourier component of the exchange interaction).     

Critical exponents from power spectra

We have simulated the two- and three-dimensional Ising models at their respective critical points with a conventional Monte Carlo algorithm. From the power spectrum of the magnetization autocorrelations we have determined the dynamic critical exponents and obtained the valuesz = 2.16–2.19 andz = 2.05, in agreement with the results quoted in the literature. We have also studied the power spectrum for the Kardar-Parisi-Zhang and Sun-Guo-Grant equations describing interface dynamics. Arguments similar to what was recently used to conclude thatz = 4 - for model B in critical dynamics were applied to the Sun-Guo-Grant growth model and the known exact values for the roughening and dynamic exponents were obtained. From an analysis of the corresponding power spectrum in self-organized critical sand models one obtains a recently proposed hyperscaling relation.