On the extremality of the disordered state for the Ising model on the Bethe lattice

We give a simple proof that the limit Ising Gibbs measure with free boundary conditions on the Bethe lattice with the forward branching ratio k2 is extremal if and only if is less or equal to the spin glass transition value, given by tanh( _c ^SG = 1/k.The work was partially supported by the NSF grant DMS 9504513.     

Diffusive–Ballistic Transition in Random Walks with Long-Range Self-Repulsion

We prove that a class of random walks on with long-range self-repulsive interactions have a diffusive-ballistic phase transition.      

Conservation laws in a directed sandpile model

We derive a complete set of conservation laws for a directed sandpile model originally studied by Dhar and Ramaswamy, and discuss the rather minimal effect of these laws on the dynamics of avalanches.     

Exact solution of a vertex model ind dimensions

We consider a class of vertex models describing directed lines on a lattice in arbitraryd dimensions, and solve the model exactly for the Cartesian lattice and in the case that each loop of lines carries a fugacity - 1. Our analysis, which can be carried out for arbitrary lattices, is based on an equivalence of the vertex model with a dimer problem. The dimer problem is, in turn, solved using the method of Pfaffians. It is found that the system is frozen below a critical temperatureT _cwith the critical exponent = (3 –d)/2.     

Rigorous bounds for critical temperatures of O(N) spin models by transformations of block spin type

We obtain new upper bounds of critical temperatures of N-vector (Heisenberg) models. We apply a transformation of block spin type to random walk representations of the spin models, which was developed by Fröhlich et al. more than a decade ago. Though the transformation is applied just one time, the upper bounds are considerably improved.     

A planar model for interacting fermions

One-particle Green function in the paramagnetic phase of a model of interacting fermions is obtained in the planar approximation. The model is zero-dimensional, in that thermal fluctuations are the only source of kinetic energy.     

Upper bound on the free energy of the spin 1/2 Heisenberg ferromagnet

The authors obtain an upper bound on the free energy of the spin 1/2 Heisenberg ferromagnet. The zero field bound is, at low temperature, similar to the formula given by the magnon approximation. That is, its functional dependence on temperature is the same but the constant is different.Work partially supported by U.S. National Science Foundation grants DMS 9196047 (JGC) and DMS 9002416 (JPS).     

Long-range spatial correlations in a simple diffusion model

A simple anisotropic diffusion model, according to semiphenomenological arguments, exhibits long-ranged spatial correlations in uniform stationary states.     

Long-range order in the XXZ model

The existence of long-range order is proved under certain conditions for the antiferromagnetic quantum spin system with anisotropic interactions (XXZ model) on the simple cubic or the square lattice. In three dimensions (the simple cubic lattice), finite long-range order exists at sufficiently low temperatures for any anisotropy(0) ifS1, and for 0<0.29 (XY-like) or>1.19 (Ising-like) ifS=1/2. In two dimensions (the square lattice), ground-state long-range order exists under the following conditions: for any anisotropy (0) ifS3/2; 0<0.032 (XY-like) or 0.67<<1.34 (almost isotropic) or>1.80 (Ising-like) ifS=1;>1.93 (Ising-like) ifS=1/2. We conjecture that the two-dimensional spin-1/2XY model (=0) has finite ground-state long-range order. Numerical evidence supporting this conjecture is given.     

Improved lower bound on the thermodynamic pressure of the spin 1/2 Heisenberg ferromagnet

We introduce a new stochastic representation of the partition function of the spin 1/2 Heisenberg ferromagnet. We express some of the relevant thermodynamic quantities in terms of expectations of functionals of so-called random stirrings on ^d . By use of this representation, we improve the lower bound on the pressure given by Conlon and Solovej inLett. Math. Phys. 23, 223–231 (1991).Work supported by the Hungarian National Foundation for Scientific Research, grant No. 1902.     

Finite energy sectors of the XY model

Continuing the earlier work on soliton sectors, we determine all finite energy representations of the XY model for almost all parameter values. In the interior of unique vacuum regions of parameters (i.e. the large external magnetic field region ||>1), the unique irreducible vacuum representation is the only finite energy representation.At the critical values of the parameters (||=1 as well as theXY symmetric case =0, ||1), there is an infinite number of mutually nonequivalent irreducible finite energy representations. Apart from the unique irreducible ground state representation and another associated irreducible representation, these infinite number of representations arise from an infinite number of nearly zero energy excitations of the ground state with a finite total energy and may be called infrared representations.In the remaining cases, as have been studied earlier, there are two additional irreducible finite energy representations besides two irreducible ground state representations and they are topological soliton sectors with different ground state limits in positive and negative spatial infinity. (For two exceptional values of parameters (, )=(0, ±1), they also become ground state representations.)     

On laplace integrals and transfer operators in large dimension: Examples in the nonconvex case

This Letter gives detailed proofs concerning the analysis of the pair correlations for a nonconvex model. Using the transfer matrix approach, the problem is reduced to the analysis of the spectral properties of this transfer operator. Although the problem is similar to the semiclassical study of the Kac operator presented in a paper with M. Brunaud, which was devoted to the study of exp-(v/2) exp h ² exp-(v/2) for h small, new features appear for the model exp-(v/2h) exp h exp-(v/2h). Our principal results concern the splitting of this operator between the two largest eigenvalues. We give an upper and a lower bound for this splitting in the semi-classical regime. As a corollary, we get good control of the decay of the pair correlation. Some of the results were announced in a previous paper. Related WKB constructions will be developed in a later paper.Inspired by papers by M. Kac [15, 16].     

Algebras in higher-dimensional statistical mechanics - the exceptional partition (mean field) algebras

We determine the structure of the partition algebraP _n(Q) (a generalized Temperley-Lieb algebra) for specific values ofQ , focusing on the quotient which gives rise to the partition function ofn siteQ-state Potts models (in the continuousQ formulation) in arbitrarily high lattice dimensions (the mean field case). The algebra is nonsemisimple iffQ is a nonnegative integer less than 2n-1. We determine the dimension of the key irreducible representation in every specialization.Work supported by the Packard Foundation.     

Thermodynamics of interacting fermions in atomic traps

We calculate the entropy in a trapped, resonantly interacting Fermi gas as a function of temperature for a wide range of magnetic fields between the BCS and Bose-Einstein condensation end points. This provides a basis for the important technique of adiabatic sweep thermometry and serves to characterize quantitatively the evolution and nature of the excitations of the gas. The results are then used to calibrate the temperature in several ground breaking experiments on (6)Li and (40)K.     

Correlations and conductivity of interacting fermions in a disordered chain

A one dimensional system of interacting fermions in a random potential is investigated within a Hartree-Fock approximation. The model in consideration combines the onsite interaction of the Hubbard model and the stochastic site energies of the Anderson model. For a system of 120 sites and several values of the model parameters the equal time correlation functions for the charge and the spin density as well as the low frequency conductivity are calculated numerically. The gross features of the conductivity thus obtained agree with published results obtained in a Monte Carlo simulation, but there are also interesting differences.