The boson gas on a Cayley tree

We analyze the free boson gas on a Cayley tree using two alternative methods. The spectrum of the lattice Laplacian on a finite tree is obtained using a direct iterative method for solving the associated characteristic equation and also using a random walk representation for the corresponding fermion lattice gas. The existence of the thermodynamic limit for the pressure of the boson lattice gas is proven and it is shown that the model exhibits boson condensation into the ground state. The random walk representation is also used to derive an expression for the Bethe approximation to the infinite-volume spectrum. This spectrum turns out to be continuous instead of a dense point spectrum, but there is still boson condensation in this approximation.     

Entanglement of a two-particle Gaussian state interacting with a heat bath

The effect of a thermal reservoir is investigated on a bipartite Gaussian state. We derive a pre-Lindblad master equation in the non-rotating wave approximation for the system. We then solve the master equation for a bipartite harmonic oscillator Hamiltonian with entangled initial state. We show that for strong damping the loss of entanglement is the same as for freely evolving particles. However, if the damping is small, the entanglement is shown to oscillate and eventually tend to a constant non-zero value.     

Renormalization group analysis of a simple hierarchical fermion model

A simple hierarchical fermion model is constructed which gives rise to an exact renormalization transformation in a 2-dimensional, parameter space. The behaviour of this transformation is studied. It has two hyperbolic fixed points for which the existence of aglobal critical line is proven. The asymptotic behaviour of the transformation is used to prove the existence of the thermodynamic limit in a certain domain in parameter space. Also the existence of a continuum limit for these theories is investigated using informatioin about the asymptotic renomralization behaviour. It turns out that the trivial fixed point gives rise to a twoparameter family of continuum limits corresponding to that part of parameter space where the renormalization trajectories originate at this fixed point. Although the model is not very realistic it serves as a simple example of the application of the renormalization group to proving the existence of the thermodynamic limit and the continuum limit of lattice models. Moreover, it illustrates possible complications that can arise in global renormalization group behaviour, and that might also be present in other models where no global analysis of the renormalization transformation has yet been achieved.A part of the material here presented was used in the author's thesis     

Long Cycles in a Perturbed Mean Field Model of a Boson Gas

In this paper we give a precise mathematical formulation of the relation between Bose condensation and long cycles and prove its validity for the perturbed mean field model of a Bose gas. We decompose the total density ρ=ρ_short+ρ_long into the number density of particles belonging to cycles of finite length (ρ_short) and to infinitely long cycles (ρ_long) in the thermodynamic limit. For this model we prove that when there is Bose condensation, ρ_long is different from zero and identical to the condensate density. This is achieved through an application of the theory of large deviations. We discuss the possible equivalence of ρ_long≠ 0 with off-diagonal long range order and winding paths that occur in the path integral representation of the Bose gas     

A perturbed mean field model of an interacting boson gas and the large deviation principle

This is a study of the equilibrium thermodynamics of a mean-field model of an interacting boson gas perturbed by a term quadratic in the occupation numbers of the free-gas energy-levels. We prove the existence of the pressure in the thermodynamic limit. We obtain also a variational formula for the pressure; this enables us to compare the effect of a smooth quadratic perturbation with that of a singular one (the Huang-Yang-Luttinger model). The proof uses a large deviation result for the occupation measure of the free boson gas which is of independent interest.     

Orthogonality and completeness of the Bethe Ansatz eigenstates of the nonlinear Schroedinger model

A rigorous proof is given of the orthogonality and the completeness of the Bethe Ansatz eigenstates of theN-body Hamiltonian of the nonlinear Schroedinger model on a finite interval. The completeness proof is based on ideas of C.N. Yang and C.P. Yang, but their continuity argument at infinite coupling is replaced by operator monotonicity at zero coupling. The orthogonality proof uses the algebraic Bethe Ansatz method or inverse scattering method applied to a lattice approximation introduced by Izergin and Korepin. The latter model is defined in terms of monodromy matrices without writing down an explicit Hamiltonian. It is shown that the eigenfunctions of the transfer matrices for this model converge to the Bethe Ansatz eigenstates of the nonlinear Schroedinger model.     

Non-Gibbsian limit for large-block majority-spin transformations

We generalize a result of Lebowitz and Maes, that projections of massless Gaussian measures onto Ising spin configurations are non-Gibbs measures. This result provides the first evidence for the existence of singularities in majority-spin transformations of critical models. Indeed, under the assumption of the folk theorem that an average-block-spin transformation applied to a critical Ising model in 5 or more dimensions converges to a Gaussian fixed point, we show that the limit of a sequence of majority-spin transformations with increasing block size applied to a critical Ising model is a measure that is not of Gibbsian type.     

The full diagonal model of a Bose gas

This paper is the final one in a series in which we investigate some models of an interacting Bose gas using Varadhan's large deviation version of Laplacian asymptotics; in it we study the equilibrium thermodynamics of the full diagonal model of a Bose gas. We obtain a formula expressing the pressure, in the thermodynamic limit, as the supremum of a functional over the space of positive bounded measures. We analyse this formula for a large class of interaction kernels and show that there is a critical temperature below which there is Bose-Einstein condensation.     

Finite-Time Current Probabilities in the Asymmetric Exclusion Process on a Ring

We calculate the time-dependent probability distribution of current through a selected bond in the totally asymmetric exclusion process with periodic boundary conditions. We derive a general formula for the probability that the integrated current exceeds a given value N at the moment of time t. The formula is written in a form of a contour integral of a determinant of a Toeplitz matrix. Transforming the determinant expression, we obtain a generalization of the known formula derived by Johansson for the infinite one-dimensional lattice. To check the general formula, we consider the specific case corresponding to the probability of a minimal non-zero current. For this case we get an explicit analytical expression and analyze its asymptotics.