Table of contents

The authors study the averaged Green function of a tight-binding-model with exponentially distributed disorder as an explicit example for a resurgent function in the sense of Ecalle, (1981). As an application, one can find the (exact) asymptotic decay of the density of states and the large-order behaviour of the coefficients of the perturbation expansion.

A simple approximation formula, valid to the high degree of accuracy of the Toronto approximation, is given for both the off-diagonal and diagonal matrix elements of the kappa matrices which transform the bosonic realisations of the symplectic algebra into an orthonormal basis for a unitary irreducible representation of Sp(6, R).

The theory underlying Cartan's method of equivalence is outlined and applied to second-order equation fields. In particular, the technique is used to obtain the canonical horizontal distribution associated to a second-order equation field-a construction which has been presented before but in a rather ad hoc manner.

A slightly more general orthogonality relation for the Hahn polynomials of a continuous variable than the recent one given by Atakishiyev and Suslov, (1985), is given here.

For the discrete-time quadratic map x_t+1=4x_t(1-x_t) the evolution equation for a class of non-uniform initial densities is obtained. It is shown that in the t to infinity limit all of them approach the invariant density for the map.

A computationally efficient form of the Hill determinant method is devised, giving both E and (x²) values, and is used to test the accuracy of that method in the light of recent criticisms. l^-1 and N^-1 asymptotic theories are used to check the numerical results.

The solutions of the equations of motion for the level population and photon number operators are obtained. The characteristic and photon distribution functions, the statistical moments of photon numbers and the correlations of modes are found.

Biased diffusion on a segment with a dangling end is considered, employing a method previously developed by the authors. It is shown that a definition of the drift velocity which is different from that of White and Barma (1984) leads to non-singular behaviour as a function of bias. The two approaches to biased diffusion are compared and a critique of their validity is presented.

The authors propose a diffusion equation for transport on a loopless aggregate. The relevant distance variable is the chemical distance and the spatially dependent coefficients of the diffusion equation are found in terms of B(l), the number of bonds at chemical distance l from the vertex of the tree, and B^s(l), the number of bonds on the skeleton of the tree. It is shown that the resulting motion can be modelled in terms of a biased one-dimensional random walk with a probability of remaining in place that approaches 1 as l to infinity . This is interpreted as being due to motion along dead ends. An important consequence of the model is a simple derivation of the expressions relating the diffusion exponent and fracton dimension to the fractal dimension and the intrinsic dimensions of the tree and its skeleton. The mean first-passage time to go from the vertex to an arbitrary shell is also found.

Two new models of three-dimensional anisotropic spiral self-avoiding walks are introduced with different types of spiral constraint. Series expansions for the two models are derived and analysed. One model is found to behave like the isotropic three-dimensional self-avoiding walk, while the other model appears to belong to a distinct universality class, with exponents nu approximately=0.655 and gamma approximately=1.24. It is argued that for these non-Markovian, undirected, unweighted walks, the absence of a plane of reflection symmetry in the allowed walks signals a new universality class.

The electrostatic interaction between charged, colloidal particles trapped at an air-water interface is considered using linearised Poisson-Boltzmann results for point particles. In addition to the expected screened-Coulomb contribution, which decays exponentially, an algebraic dipole-dipole interaction occurs that may account for long-range interactions in interfacial colloidal systems.

Stationary state solutions with finite energy are presented for generic homogeneous nonlinear Dirac equations in (1+1) dimensions. These models fall into two distinct classes, with dual existence conditions. In one case the soliton solution is unique and extends throughout the infinite line R¹ while for the second class the soliton exists on a finite interval, and it is not unique.

General SU(2) Yang-Mills configurations and solutions of the source-free Yang-Mills equations are investigated, which are invariant under some high subgroup of the Poincare group. Simply transitive G₃, all multiply transitive G₃, multiply transitive G₄, which can be obtained by adding a fourth commuting Killing vector to a multiply transitive G₃, and also all groups G_r, r>or=6 are considered. The results presented mainly concern the question of existence. Selected special invariant configurations are given explicitly.

For an arbitrary real semisimple Lie algebra g, a method is presented of constructing skew-Hermitian realisations in terms of quantum canonical variables, p_i, q_i. The construction starts with a decomposition g=n₊^b(+)g₀^b(+)n_-^b of g, which is a simple generalisation of the triangle decomposition, employing substantially an induced representation of g with respect to a suitable representation sigma of the subalgebra g₀^b(+)n_-^b. It is shown that the realisations obtained are Schurean. As examples, the realisations for the algebras sp(2, R) and gl(3, R) are calculated explicitly.

Generalised characters of the infinite dimensional, holomorphic, discrete series, unitary, irreducible representations of the non-compact groups U(p, q), Sp(2n, R) and SO*(2n) are explicitly expressed in terms of characters of finite dimensional unitary group representations. These formulae are remarkably succinct despite involving certain infinite series of Schur functions. Similar formulae are derived for harmonic series unitary representation of both U(p, q) and Sp(2n, R). Consideration of the branching rules from U(p, q) to U(q)*(p) and from Sp(2n, R) to U(n) enables holomorphic representations to be identified as a subset of the harmonic representations. The branching rules are established in full generality and are then used in the evaluation of tensor products of both holomorphic and harmonic representations. In the case of the former a known result is recast in terms of closed formulae involving Schur functions and for the latter various generalisations of these formulae are given. A conjecture is also made regarding what might be the simplest possible formulae covering all holomorphic and harmonic representations of Sp(2n, R) and U(p, q). Illustrative examples are presented.

The symmetric Fock space for a given representation of a compact internal symmetry group is shown to be naturally associated with an induced representation space, induced by a unitary group. The induced representation space is a Hilbert space over the complex sphere, and the orbits on this complex sphere with respect to the internal symmetry groups give rise to a double coset decomposition. A scattering operator acting on double coset representatives only is shown to be unitary, invariant with respect to the internal symmetry group, and include particle production. A simple example in which positively and negatively charged mesons generate a Fock space with respect to a U(1) internal symmetry group is given.

The authors discuss supersymmetric regularisations of soluble supersymmetric quantum mechanical models with continuous parts to the energy spectra. Results include a demonstration that the Witten index changes discontinuously from zero to one upon removal of the regularisation.

A classical pseudo-spin model in one dimension is considered, representing a variation on the Frenkel-Kontorova model to include non-convex interactions, resulting in three competing length scales. An exact algorithm is used numerically to determine the classical ground state as a function of a chemical potential. Approximate arguments suggest that only a first-order transition should occur. However, when the lengths are not rationally related, it is found that the mean lattice spacing appears to vary as a devil's staircase. The plateaux of the staircase correspond to locking to incommensurate structures while there is no locking to the commensurate ones. Most of the locking values observed numerically belong to a series which can be analytically calculated on the basis of simple topological hypotheses which are also consistent with numerical observations.

Self-avoiding walks on the square lattice with nearest-neighbour attractive interactions are investigated as a model for the two-dimensional version of a polymer in dilute solution. Temperature dependences of the exponent nu and the free energy of the chain are estimated from the exact enumeration data for up to 20 steps; the value of nu at the theta -point disagrees with the mean-field theory. The end-distance distribution function at the theta -point is also examined.

For pt.I see ibid., vol.18, p.2479, (1985). A quantum Monte Carlo method developed in the first paper of this series is used to investigate the spin-1/2 isotropic ferromagnet and antiferromagnet. Critical indices are computed in a high precision simulation. The authors obtain alpha =-0.261+or-0.013, gamma =1.552+or-0.008 for the ferromagnet, and alpha =-1.202+or-0.009, gamma =1.132+or-0.012 for the antiferromagnet.

The distribution function ( sigma sigma ) in the inhomogeneous interface zone modelled by a two-dimensional SOS (solid-on-solid) system is successfully represented as polynomials in local density and the lattice analogues of its gradients and higher derivatives.

A three-state IRF model of a magnetic hard-square lattice gas with anisotropic interactions is solved exactly on special two-dimensional manifolds in the full five-dimensional thermodynamic space spanned by the (ferro- or antiferro-) magnetic interactions J, K, the (attractive or repulsive) lattice-gas interactions L, M and the activity z. The phase boundaries determined include multicritical lines and critical surfaces as well as first-order surfaces of three- and fivefold phase coexistence between the fluid and various (para, ferro, etc.) magnetic square-ordered solid phases. Analytic expressions are given for free energies, interfacial tensions, correlation lengths, sublattice densities and magnetisations. The critical behaviour of these quantities is studied and the associated critical exponents obtained.

An isomorphism between the Gaussian model in the magnetic field and the time-dependent conduction-diffusion problem permits the author to develop a real space renormalisation group method for the diluted version of this model by which all single cluster results of the phenomenological scaling theory of Gefen, Aharony and Alexander (1983) are correctly reproduced. An extension of this model to the case of strong bias then provides some restrictions on the validity of their results. The result for the density of states, which the author reproduces in the form predicted by Alexander and Orbach, (1982) is subject to the same kind of restrictions.

The author proposes that the following relations hold exactly for the lattice animal model of branched polymers. The resistivity exponent zeta _a is exactly equal to d_a, the fractal dimensionality of the backbone of the anomals. The random walk fractal dimensionality d_w is given by, d_w=d_a+d_a, where d_a is the fractal dimensionality of the animals. The spectral dimension d_s of the backbone of the animals is given by, d_s=1 at all dimensions, i.e. the backbone is a highly decorated but essentially chain-like object. These are in contrast with earlier suggestions that one needs two (three) topological properties to obtain zeta _a(d_w). The author proposes that these relations may hold for all clusters for which loops are irrelevant and have a finite upper critical dimensionality. The implications of these results for diffusion-limited aggregates and percolation clusters are also discussed.

The authors provide a perturbative check of the Ward-Takahashi identities of QED at finite temperature using a new real time formalism, thermo field dynamics. Gauge invariance is shown to hold up to the two-loop level except one special case. In verifying the identities they find a consistent rule in dealing with products of singular terms which are characteristic of the real time formalism.

The Hamiltonian H(x,p)=1/2(p₁²)+1/2(p₂²)+1/2(x₁²x₂²) is investigated. The different aspects (singular point analysis, stability analysis, numerical treatment etc.) for studying chaotic behaviour are discussed. Moreover, the authors discuss the quantised version of the Hamiltonian.

The authors provide the first description of the surface transition that goes beyond mean-field theory and that develops a systematic epsilon -expansion method for this transition. The theory exhibits the emergence of (d-1)-dimensional physics with a complicated dependence on c (the surface interaction parameter). A calculation is provided for the lowest correction to the shift in the critical temperature (from its mean-field value).

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