Mean-field bounds and correlation inequalities

I prove a new correlation inequality for a class ofN-component classical ferromagnets (1N4). This inequality implies that the correlation functions decay exponentially and the spontaneous magnetization is zero, above the mean-field critical temperature.     

Symmetries and correlation inequalities for classicaln-vector models

We describe a new class of single spin measures on then-dimensional sphereS _r ⁿ of radiusr (n 4) for which Lebowitz-type [J. Lebowitz,J. Stat. Phys. 16:463 (1977)] inequalities hold. This is achieved by an appropriate parametrization ofS _r ⁿ . The above class includes the uniform measures onxs Rⁿ ¦x¦ r for any 0 p r. The second topic of this paper is an abstract formulation of the first Griffiths inequality [R. B. Griffiths,J. Math. Phys. 8:478 (1967)] and the underlying symmetry property.     

Some correlation inequalities for Ising antiferromagnets

We prove some inequalities for two-point correlations of Ising antiferromagnets and derive inequalities relating correlations of ferromagnets to correlations of antiferromagnets whose interactions and field strengths have equal magnitudes. The proofs are based on the method of duplicate spin variables introduced by J. Percus and used by several authors to derive correlation inequalities for Ising ferromagnets.     

Intersite correlation in anisotropic ferromagnetic spin one systems

The effect of intersite correlations, in Callen approximation, on the critical temperature T_C has been investigated for the anisotropic spin one system in the presence of uniaxial anisotropy. The effect is found to be important at small anisotropy.     

On two correlation inequalities for Potts models

The Fortuin-Kasteleyn random cluster representation ofq-state Potts models is used to extend to everyq two correlation inequalities proven previously only for even values ofq.     

Zeros of partition functions via correlation inequalities

We show analyticity of the pressure for some classical ferromagnetic systems in the region ¦Im ¦ < Re of the external field. The proof, via correlation inequalities, is simpler than existing proofs for the Lee and Yang region {Re 0} and applies, without any approximation procedure, to more general continuous spin variables, e.g., distributed as , where _2n is an arbitrary real number and the other parameters are positive. It also applies directly to plane rotators in the region ¦Im ¦ ¦Re ¦ (Euclidean norms), but the proof will be given in a subsequent article, together with new inequalities between truncated correlation functions.     

Correlation inequalities and uniqueness of the equilibrium state for the plane rotator ferromagnetic model

We derive new inequalities for the plane rotator ferromagnetic model and use them to obtain the following results:

1. If the model is isotropic, the derivability of the free energy as function of the magnetic fieldh implies the existence of a unique translation invariant Gibbs state and if furthermoreh=0 all Gibbs states are invariant by rotation of the spins.
2. If the model is anisotropic the above assertion holds forh non-zero.
3. If the model is anisotropic then there are at most two extremal translation invariant Gibbs states for almost all values of the anisotropy parameter.

     

The random-walk representation of classical spin systems and correlation inequalities

We use the random-walk representation to prove the first few of a new family of correlation inequalities for ferromagnetic ?⁴ lattice models. These inequalities state that the finite partial sums of the propagator-resummed perturbation expansion for the 4-point function form an alternating set of rigorous upper and lower bounds for the exact 4-point function. Generalizations to 2n-point functions are also given. A simple construction of the continuum ? _d ⁴ quantum field theory (d<4), based on these inequalities, is described in a companion paper.     

More correlation inequalities for a class of even ferromagnets

Rigorous correlation inequalities are presented for a class of even ferromagnets, which includes the spin-1/2 Ising model and scalar ⁴ models. One of them leads to an extension of the Glimm and Jaffe uniform upper bound on the ⁴ renormalized coupling constant into the nonsymmetric regime.     

Applications of correlation inequalities to low density graphical codes

This contribution is based on the contents of a talk delivered at the Next-SigmaPhi conference held in Crete in August 2005. It is adressed to an audience of physicists with diverse horizons and does not assume any background in communications theory. Capacity approaching error correcting codes for channel communication known as Low Density Parity Check (LDPC) codes have attracted considerable attention from coding theorists in the last decade. Surprisingly strong connections with the theory of diluted spin glasses have been discovered. In this work we elucidate one new connection, namely that a class of correlation inequalities valid for Gaussian spin glasses can be applied to the theoretical analysis of LDPC codes. This allows for a rigorous comparison between the so called (optimal) maximum a posteriori and the computationaly efficient belief propagation decoders. The main ideas of the proofs are explained and we refer to recent works for the more lengthy technical details.     

Scalar quantum electrodynamics on lattice correlation inequalities and infinite volume limit

We consider Wilson's lattice approximation to scalar Quantum Electrodynamics. We establish correlation inequalities of Griffiths-Kelly-Sherman type and a rigorous transfer matrix formalism, from which existence and non triviality of the infinite volume limit follow.     

The random walk representation of classical spin systems and correlation inequalities

Ferromagnetic lattice spin systems can be expressed as gases of random walks interacting via a soft core repulsion. By using a mixed spinrandom walk representation we present a unified approach to many recently established correlation inequalities. As an application of these inequalities we obtain a simple proof of the mass gap for the (⁴)₂ quantum field model. We also establish new upper bounds on critical temperatures.Partially supported by N.S.F. Grant No. 79-02490On leave from the University of VirginiaPartially supported by N.S.F. Grant No. DMR 81-00417     

Hund's rules reformulated as inequalities and some related inequalities for exchange integrals,Slater-Condon parameters energy differences and correlation energies

Measurements have been made of the neutron scattering structure factors of liquid nitrogen and liquid oxygen at 77 K and 84 K respectively in the Q-value range of 3 to 36 Å^-1. ‘White’ incident thermal neutrons were produced in the wavelength range of 0·3 to 3·0 Å by a pulsed electron linac and detected in a total-scattering time-of-flight spectrometer. Qualitative agreement has been obtained between these present data and a simple molecular form factor in the Q-value range of 12–36 Å^-1.

Using reactor data [1], structure factors over a total Q-value range of 0·3 to 36 Å^-1 were obtained. Fourier transforms of the structure factor curves yield pair distribution functions which show a distinct separation of the inter and intra-molecular distances in the liquid. A comparison is made with results for the solid phase.     

The GHS and other correlation inequalities for a class of even ferromagnets

We prove the GHS inequality for families of random variables which arise in certain ferromagnetic models of statistical mechanics and quantum field theory. These include spin –1/2 Ising models, ⁴ field theories, and other continuous spin models. The proofs are based on the properties of a classG ^– of probability measures which contains all measures of the form const exp(–V(x))dx, whereV is even and continuously differentiable anddV/dx is convex on [0, ). A new proof of the GKS inequalities using similar ideas is also given.Supported in part by National Science Foundation Grant MPS 71-02838 A 04.Supported by National Science Foundation Grant MPS 74-24696.Supported in part by National Science Foundation Grant MPS 74-04870.     

Exact analytic formulae for the correlation times for single domain ferromagnetic particles

Exact solutions for the longitudinal relaxation time T_∥ and the complex susceptibility χ_∥(ω) of a thermally agitated single domain ferromagnetic particle are presented for the simple uniaxial (Maier-Saupe) potential of the crystalline anisotropy considered by Brown [Phys. Rev. 130 (1963) 1677].     

Remarks on the relation between the Lee-Yang circle theorem and the correlation inequalities

We investigate the relation between the Lee-Yang circle theorem and the correlation inequalities. These results are general and independent of models. General properties of the partition functions which belong to the Lee-Yang class are given.     

A study of perfect wetting for Potts and Blume-Capel models with correlation inequalities

The so-called perfect wetting phenomenon is studied for theq-state,d2 Potts model. Using a new correlation inequality, a general inequality is established for the surface tension between ordered phases ( ^a,b ) and the surface tension between an ordered and the disordered phases ( ^a,f ) for any even value ofq. This result implies in particular at the transition point _t where the previous phases coexist forq large. This inequality is connected to perfect wetting at the transition point using thermodynamic considerations. The same kinds of results are derived for the Blume-Capel model.     

On a class of inequalities and identities for spin correlation functions of general Ising models

A recently obtained result concerning a class of inequalities and identities for spin correlation functions of Ising models with pair interactions is extended to general Ising models.     

Properties of then-body correlation functions near the liquid-gas critical point. Correlation inequalities

We investigate the behavior of the many-body correlation functions in the vicinity of the gas-liquid critical point. We use the framework of the liquid state theory and, accordingly, no reference to an effective Landau-Ginzburg Hamiltonian is made. The critical condition is introduced by means of the equation of state. From the Baxter equation relating the many-body correlation functionsh(n) andh(n+1), we find that the integrals of all theh(n) diverge at the critical point. Then we present strong arguments and this leads to GKS-like inequalities, under some limiting conditions: the interparticle distances must be large and the thermodynamic state of the system must be close to the critical point. In order to get these inequalities, an upper bound forh(n) is obtained. Particular attention must be paid to the fact that the usual asymptotic approximations of the liquid state theory are no longer valid.