Table of contents

For pt.I see ibid., vol.19, no.3 p.321-7 (1986). Modification rules are presented for all finite-dimensional, atypical representations of OSp(M/N). With the representations specified by Young supertableaux, these rules relate non-standard supertableaux to standard supertableaux via the removal of continuous boundary hooks.

The vector differential equation Del *a=ka with k constant appears in models of 'force-free magnetic fields' in plasma physics and in astrophysics, and also in a model for 'force-free electromagnetic waves'. Solutions of this equation which are identified with physical fields exhibit several inconsistencies. This letter identifies the source of these inconsistencies as the lack of covariance of this equation with respect to transformations.

The notions of Lie and conformal invariance symmetry for a regular second-order equation field are shown to be essentially identical. It is pointed out that the way in which conformal invariance symmetries are traditionally defined is not invariant and that this leads to difficulties in interpreting a recent result of Logan (1985). However, the introduction of coordinate-free geometrical apparatus leads to a simple proof of Logan's result and interpretation of the examples considered by him.

The authors write explicit integral expressions for sums of inverse powers of the eigenvalues of the Laplacian with Dirichlet boundary conditions in a simply connected bounded two-dimensional domain.

Starting from the N-particle Smoluchowski equation without hydrodynamic interactions the authors derived an expression for the mean square displacement by modelling the memory function entered in the equation for the incoherent scattering function. The numerical evaluation of that expression is done for the case of spherical particles interacting via a screened Coulombic potential. The results of this theory compare favourably with computer simulation results.

The pair correlation function of a large connected cluster, formed during the irreversible aggregation of a two-dimensional distribution of colloidal microspheres, shows a distinct crossover from flocculation to gel-sol scaling behaviour consistent with kinetic clustering of cluster model predictions. Smaller isolated clusters arising out of the same aggregation even have lower than expected fractal dimensions, suggesting the influence of anisotropic interactions.

Linear diffusion-limited aggregation is studied on the square lattice. The radius of gyration exponent nu =0.79+or-0.01 is estimated using the Monte Carlo method and a scaling analysis.

Using the reaction-diffusion master equation of chemical kinetics, the authors study fluctuations and correlation functions around non-equilibrium steady states. They demonstrate that the underlying Lagrangian description in the Poisson representation possesses in the equilibrium limit an exact symmetry which ensures that statistical mechanics is recovered and that the intrinsically non-equilibrium long-range correlations in these systems vanish. An explicit form for the two-time single-particle correlation function, including both the long-range and short-range contributions, is constructed using mean-field arguments.

The local height probabilities P_a and the mean square height average (h²) are calculated exactly for the smooth phase of the body-centred solid-on-solid model. Near criticality P_a approximately K'(-t)^1/4 and (h²) approximately K"(-t)^-1/2, where t is the deviation from criticality parameter and K', K" are constants specified in the text.

The author shows how to expand critical amplitudes in inverse powers of the dimensionality d. The technique, which is rather general, is applied to the classical n-vector model, with particular emphasis on the self-avoiding walk (n=0) and Ising (n=1) cases. The existence of these expansions and of similar ones for critical points and critical constants suggests that 1/d expansions for non-universal quantities are analogous to the epsilon expansions for universal quantities.

A new type of cluster in the universality class of lattice animals is introduced based on a stochastic form of the Martin algorithm by the Monte Carlo method. Lattice animals with sizes up to 1000 sites are generated. The asymptotic radius of the gyration exponent is consistent with the value 0.64, corresponding to a fractal dimension of 1.56. An investigation of anisotropy of these animals shows that small animals are anisotropic while larger ones are not.

The real space renormalisation group theory is applied to the self-avoiding Levy flight problem with Levy exponent mu on a two-dimensional square lattice. A finite-lattice renormalisation transformation is used to derive the critical behaviour exhibited by two classes of self-avoiding Levy processes: the node-avoiding Levy flight (NALF); the path-avoiding Levy flight (PALF). It is found that the NALF and the PALF belong to the same universality class.

A projection operator which maps the q-state Potts model onto an Ising model is investigated. The case of an approximate mapping between two-spin nearest-neighbour clusters on a d-dimensional hypercubic lattice is considered. In the ferromagnetic case, first-order transitions occur for q>q_c(d)=1+exp(2K_Ic(d))>or=2, where K_Ic(d) is the critical Ising coupling. Among the results are q_c(d) to exp(2/(d-1)) as d to 1⁺, q_c(2)=3.41 and q_c(3)=2.56. The antiferromagnetic Potts model has no long range order for q>2.

The nature of the bulk energy per unit volume in magneto-electric crystals implies an unusual form of Hall effect confined to the boundary surface crystal faces.

Modification rules are presented for finite dimensional graded tensor representations of OSp(M/N). With the representations specified by Young supertableaux these rules relate non-standard supertableaux to standard supertableaux via the removal of continuous boundary hooks. All typical tensor representations are treated together with atypical representations which satisfy up to two atypically conditions.

For the uniform asymptotic expansion of certain types of contour integrals, one of whose critical points is an endpoint of the interval of integration, a method alternative to Bleistein's is introduced and numerically tested by way of a non-trivial physical example.

The author considers an infinitely long conducting cylinder whose dielectric and magnetic permittivity and conductivity are functions of the distance from a point inside the cylinder to its axis. It is shown that the r-dependent part of the set of electromagnetic modes associated with such a cylinder is complete and orthogonal, and several completeness relations are constructed, which are different from those postulated in the literature.

The exceptional sets in the autonomous and periodic KAM theorems consist of sets of near-resonant frequencies ( omega ₁, . . ., omega _n)= omega for which the method of proof of the theorems breaks down. They are residual and of measure 0. Their Hausdorff dimension is deduced from that of a related set of well approximable linear forms. The Hausdorff dimension of these linear forms is a special case of a more general number theoretic result on systems of linear forms established recently but a self-contained proof is given. The proof relies on the resonant hyperplanes being reasonably well distributed. A fairly general statistical 'second moment' argument is used but some geometrical ideas are introduced to simplify and improve the original proof. When the number of degrees of freedom is large, the Hausdorff dimension is nearly maximal, so that, although of measure 0, the exceptional sets are, roughly speaking, close to sets of positive Lebesgue measure. The implications for the stability of Hamiltonian systems with many degrees of freedom and for the onset of certain kinds of instability are discussed briefly.

Pulsed wavefields contain moving lines, called wave dislocations, where the amplitude is zero. As the lines move they sweep out surfaces called dislocation trajectories. The authors describe an experiment with ultrasound designed to test the theoretical prediction of Wright and Nye (1982) that, for small bandwidth, the trajectories are close to parts of frequency minimum surfaces. That is, surfaces on which the corresponding continuous-wave amplitude pattern has a minimum with respect to changes in frequency. 'Close' here means to second order in the bandwidth, and the prediction is indeed confirmed to this accuracy.

Using the exact generating functions for solid-on-solid models, the authors construct 'necklace' partition functions for wetting and prewetting at a sticky wall in two (bulk) dimensions. Three-component generalisations of these models, constructed to study interfacial wetting, are introduced and solved. Based on these models they discuss the types of interfacial critical behaviour to be expected in three-component systems in two dimensions.

A thermodynamic criterion for the persistence of an irreversible process is derived from the method of associative distributions applied to the probability of large deviations in stochastic processes. Paths of relative maximum likelihood are determined from the condition that the distribution and associated distribution be non-defective. The latter predicts the threshold value of the noise strength and the paths are mirror images in time of one another. Based on the analogy with the central limit theorem and the method of associated distributions, the technique is used to estimate the error in approximating the actual distribution by one that is obtained in the diffusion limit, analogous to the normal distribution, for large deviations. The maximum power of the deviation is discussed in terms of the fractal dimension.

It is shown that the partition function for the lambda -state Potts model with pair interactions is related to the expected number of integer mod- lambda flows in a percolation model. The relation is generalised to the pair correlation function. The resulting high-temperature expansion coefficients are shown to be the flow polynomials of graph theory. The authors also prove an observation of Tsallis and Levy (1981) concerning the equivalent transmissivity of a cluster.

The (2+1)D Ising model on a triangular lattice is explored for lattices of up to 5*5 sites. Finite-size scaling techniques are used to estimate the critical parameters x_c=0.2096(2), nu =0.627(4), beta =0.322(6), gamma =1.236(8). These agree with previous estimates, within errors, and with the hypotheses of universality and hyperscaling. The behaviour at low temperatures is also studied, and theoretical predictions of finite-size scaling at a first-order transition are confirmed.

Using recent results by Cardy (1984) based on conformal invariance of critical correlation functions the authors calculate universal results for scattering functions S(k), susceptibilities, correlation lengths and specific heat correction terms for finite Ising systems in two dimensions with circular, elliptical and rectangular shapes and free boundary conditions. Results show the effects of critical order on these quantities. For a circle, S(k) decays as 1/k^{2- eta} for a large range of intermediate k values with an 'apparent' exponent eta =0.09. The probable influence of end, edge and domain wall effects in the rectangular geometry is discussed. Application of the results to experimental systems and other theoretical models is discussed.

The author develops a method for studying the high-order behaviour of the perturbation expansion in theories in which the number of field components, n, is taken to zero. This procedure is illustrated on the field theory formulation of the percolation problem, which can be considered as the n=0 limit of the (n+1)-state Potts model. The saddle points controlling the asymptotic behaviour are labelled by an integer r=1, 2, . . ., n for positive integer n, but after continuation to n=0 the dominant contribution effectively comes from the saddle point with r= infinity . The perturbation expansion is found to be oscillatory at large orders and its behaviour is calculated.

By means of a commutation formula, the author gives a simple proof of the upper bound of Wong et al. on the gap between the first two eigenvalues in the Schrodinger operator. Unfortunately this proof does not seem to generalise into higher dimensions.