Table of contents

For lattice sites at integer values along the line let the probability of a bond between sites m and n be p/ mod m-n mod ^s, m not=n, 0<or=p<1, 1<s<or=2. The author proves that for p<p_c^(B)=1/2 zeta (s) there is no infinite cluster and that for s>2 there is never an infinite cluster.

A generalisation of the self-avoiding walk is introduced in which k or higher multiple points are forbidden (k=2 corresponds to the standard self-avoiding walk). The Flory theory gives the radius of gyration exponent nu _k=(k+1)/((k-1)E+2) when E<or=E_c(k)=2k/(k-1). E is the Euclidean dimension of the problem and E_c(k) the upper critical dimension which is also obtained using the fractal set theory.

A two-dimensional fractal model is constructed for diffusion controlled deposition on a surface. The fractal geometry of the deposit and the power law behaviour of the quantities characterising the non-equilibrium cluster size distribution are shown to be consequences of the competition generally present in a nonlinear growth process. A qualitative agreement with previous numerical results is found and the scaling laws for the critical exponents of the problem are shown to be satisfied exactly.

The rational approximation method is used to examine estimates for the first and second derivatives of the critical exponent of the Ising model with respect to a variable related to the spin on the three-dimensional body centred cubic lattice. A variation of the method of critical point renormalisation is used to eliminate bias due to uncertainty in the location of the critical point. The results support the concepts of universality of gamma with respect to the spin.

The relation between critical exponents and the amplitude of the correlation length divergence at the critical point of two-dimensional systems as a function of finite size is investigated and generalised in two ways. Correlations more general than those of order-order type are included. A form appropriate for anisotropic systems is proposed. The authors present (a) exact results for the Ising and Gaussian models and (b) numerical results for the symmetric eight-vertex (Baxter), continuous q-state Potts, and continuous N-component cubic models.

The authors study the effects of branching on linear polymers using a two-parameter position space renormalisation group. From the resulting phase diagram and the renormalisation group flows they find a new higher-order critical point. They propose that this critical point describes a vulcanisation process in which both the linear polymer and its branches become part of an infinite branched polymer network.

Cyclotron radiation carries a continuous flux of energy, momentum and angular momentum. Current theory cannot explain how such quantities are derived from the emitting particles (electrons).

Ground state properties of finite anisotropic antiferromagnetic Heisenberg chains are studied for odd half-integer spins S=¹/₂, ³/₂ ... as well as integer spins S=1, 2 .... Finite size scaling analysis of the results clearly distinguishes the half-integer (S=¹/₂, ³/₂) from the integer (S=1) spin situation. It gives strong support to a recent conjecture which postulates that the T=0 phase structure is very different in the two cases. According to this idea there exists a new phase between the planar and the antiferromagnetic region, for integer spins only. This phase, which includes the isotropic point, has a finite energy gap and no long range order.

A derivation of Sompolinsky's result (1981) for the long range spin glass is given starting from the set of Thouless-Anderson-Palmer (TAP) equations. No replicas are used. To describe the system at successive levels of averaging (over successive time scales or associated TAP solutions) the method introduces an infinite sequence of fluctuating local fields.

To derive selection rules for different physical processes occurring in polymers and quasi-one-dimensional solids one has to determine the reduction coefficients for the Kronecker products of the irreducible representations of their symmetry groups, the line groups. This task is accomplished here and the coefficients are tabulated explicitly for all the line groups isogonal to C_n, C_nv, C_nh and S_2n (n=1, 2, ...) point groups.

It is shown that the Lie algebra of globally Hamiltonian vector fields on a compact symplectic manifold can be lifted to a Lie algebra of smooth functions on the manifold under Poisson bracket. This implies that any algebra of symmetries of a classical mechanical system described by such a manifold may be realised as an algebra of observables (smooth functions). Parallels between lifting problems in classical and quantum mechanics are explored.

Let u'=B eta u and L be, respectively, the elementary Backlund transformation and hierarchy generating operator for the AKNS equations. It is shown that (dB/d eta )(B eta )^-1= sigma ₃/(L- eta ). A similar formula relating to the general N*N matrix spectral problem is also derived.

It is pointed out that the semiclassical periodic orbit expansion for the level density of a bounded quantum system may contain singular structure not found in the exact level density. This is illustrated for a particular system, a pseudointegrable polygon billiard. The author shows that here the periodic orbit expansion has singular structure on all scales of energy, and takes on negative as well as positive values. However, when sufficiently smoothed the expansion bears a good resemblance to the exact smoothed level density.

The classical dynamics of a billiard which is a quadratic conformal image of the unit disc is investigated. The author gives the stability analysis of major periodic orbits, present the Poincare maps, demonstrate the mixing properties by following the evolution of a small element in phase space, show the existence of homoclinic points, and calculate the Lyapunov exponent and the Kolmogorov entropy h. It turns out that the system becomes strongly chaotic (positive h) for sufficiently large deformations of the unit disc. The system shows a generic stochastic transition. The computations suggest that the system is mixing if the boundary is not convex.

The determinantal formalism previously built up in the evolution operator problem is extended to the density matrix equation in the case, here, of a Heaviside step external perturbation, and in the presence of internal collision potential. Reduced determinantal forms, matched to those first derived for the evolution operator, are obtained and compared with both the iterative expansion and the initial Fredholm-Laplace solution. They not only exhibit the multiple transition structure of the physical response, along with the associated transition width upon expansion, but also the trace conservation requirement is shown to be satisfied at any order and all times. An illustration is given in a simplified physical system.

The author describes a new algebraic treatment of the chain-model eigenvalue problem. It is based on the (2t+1)-diagonal and asymptotically smooth structure of the corresponding Hamiltonians. The formalism is illustrated on the phenomenological doubly anharmonic oscillators.

The quantum inverse scattering method is used for the study of field theories with quartic-type interaction in 1+1 dimensions. Appropriate Lax pairs are constructed, and the corresponding Heisenberg fields are recovered from the quantised scattering data.

The ground state energy of a hydrogenic atom of nuclear charge Z, perturbed by a polynomial perturbation 2 lambda Zr+2 lambda ²r², is calculated by means of a variational modification of Rayleigh-Schrodinger perturbation theory, which is effective for all negative lambda .

The method of obtaining the explicit expressions of all the even- odd-parity exact solutions of the Schrodinger equation for the interaction x²+ lambda x²/(1+gx²) is discussed when the couplings lambda and g satisfy some specific relations. In the general case a simple equation for approximating the energy eigenvalues has been developed.

Monte Carlo calculations are used to investigate some statistical properties of random walks on fractal structures. Two kinds of lattices are used: the Sierpinski gasket and the infinite percolation cluster, in two dimensions. Among other problems, the authors study: (i) the range R_N of the walker (number of distinct visited sites during N steps): average value S_N, variance sigma _N and asymptotic distribution: (ii) renewal theory (return to the original site): probability of return P₀(N), mean number of returns nu _N. The probability distribution of the walker position P(N,R) after N steps is discussed. The asymptotic behaviour (N>>1) of these quantities exhibits power laws, with associated exponents. The numerical values of these exponents are in good agreement with recent theoretical predictions (Alexander and Orbach, 1982; Rammal and Toulouse, 1982).

The existence of a massless phase in several different two-dimensional O(2)-symmetric spin models is investigated within the approximate Migdal/Kadanoff renormalisation transformation. The models investigated include the planar rotor model, the truncated model used by Nienhuis (1982) to derive exact results for O(2)-symmetric models, and the step model of Guttmann and Joyce (1973). The truncated model is found to exhibit a 'fixed' line similar to that seen in the planar rotor model, but only for values of the coupling for which the model is unphysical. The step model is found to be always disordered, but a variant exhibits a fixed line. The significance of the results beyond the Migdal approximation is assessed.

The critical behaviour of an Ising like model with three-spin coupling is studied in a transverse field. A self-dual renormalisation group transformation can be applied which gives well converging results for the critical exponents when large enough cells are taken. Finite size scaling on the same model give comparable results.

The loop gas in d=2, 3, 4 and 5 dimensions and with multiplicities m=0, 1, 1.5, 2, 3 and 4 is investigated by the Monte Carlo method. The critical temperatures and approximate values for the critical exponent nu corresponding to second-order phase transitions are obtained for d=2 and 3.

Critical spin wave dynamics in the dilute Heisenberg chain near the percolation threshold is treated by two complementary approaches. The first exploits the renormalisation group transformation of parameters under a length scaling achieved by decimation on the equations of motion of the random system, the second obtains from the contributions from chain segments of all possible sizes the average dynamic response using a continuum approach valid for small wavevector k and long percolation correlation length xi _p. Both approaches yield identical dynamic scaling forms for the dynamic response, with dynamic exponent z=2, and details of the crossover of characteristic frequency between hydrodynamic and critical form as k xi _p varies from 0 to infinity . A detailed expression for the scaling function for the dynamic response is also obtained.

The authors construct dimensionless actions for fields which have non-trivial central charge behaviour. By means of optimisation methods for systems with constraints, they show how such actions must be defined in order to give the correct four-dimensional equations of motion. This is considered initially for a single real scalar free field with one central charge, and extended to successively more complicated situations. In particular they discuss fields with more than one central charge with suitable constraint conditions. They also consider actions for superfields with central charges and obtain full superspace actions for the degenerate cases. Fields without central charges are also described by integration over central charge dimensions. They conclude that their four-dimensional world may be embedded as the boundary of a high-dimensional manifold whose extra dimensions are not directly accessible on-shell.

The authors decompose any spin-reducing representation of N=2 supersymmetry into a direct integral of irreducible representations in a unique manner, in which each irrep has standard form. A single real number is required to classify the irreps in the general representations.

An elementary proof of the distributional epsilon to =0 limit of renormalised (absolutely convergent for epsilon >0 in momentum space) Feynman amplitudes is given which makes use of simple and direct means and establishes the convergence of the amplitudes in Minkowski space.

The spectrum of elementary excitations for scalar field theories on a lattice is calculated in the large-N approximation and is found to consist of two distinct branches, one of which may be interpreted as a collective mode. The method of calculation is based on a generalised Holstein-Primakoff theory for the real symplectic algebra.

Kahler (1962) has used the correspondence between the exterior and Clifford algebras to formulate a version of the Dirac equation (the Kahler equation). The authors exploit the fact that exterior forms may thus be used for the description of half-integer as well as integer spin to present a supersymmetric model.

The authors study the static property of semi-dilute polymer solutions in a good solvent. Edwards' transformation is employed to develop the conformation space renormalisation group theory in semi-dilute solutions. They are primarily concerned with the crossover behaviour from a dilute to semi-dilute regime. The first-order calculation on epsilon =(4-d) with d the dimensionality of space is performed for the asymptotic scaling functions of the radius of gyration, osmotic compressibility and the correlation length of the monomer density fluctuations.

The normal ordering formula for (N+a^r)ⁿ, where N=a⁺a, is derived for arbitrary r and n. This is a generalisation of the results recently obtained by Mikhailov (1983) for r=1, 2. Some other closely related results are also discussed.