Table of contents

It is shown that maximum entropy distributions having prescribed the finite number of moments can be determined from a suitable system of differential equations.

A quasi-crystalline solid is an aperiodic array of atoms which gives rise to Bragg-like diffraction peaks. Using the projection method for generating such structures, the authors provide a simple derivation of this paradoxical behaviour.

The construction of the (Euclidean) quantum mechanics on a manifold are generalised to two-dimensional models of quantum field theory. Partial differential equations for Euclidean fields are obtained, which are random perturbations of instanton equations.

The authors present the results of a finite size analysis of the one-dimensional spin-1/2 dimerised Heisenberg chain. The value of the gap exponent, nu , is obtained using two extrapolation procedures. When logarithmic corrections are taken into account, they obtain the values nu =0.668+or-0.001 and nu =0.667+or-0.002, in good agreement with the conjectured value nu =2/3. They compare these results with those obtained assuming a pure power-law behaviour.

A differential equation method is used to derive a closed form expression for the double Laplace transform of the l wave reduced Coulomb Green function in terms of Gaussian hypergeometric functions. The expression is found to reproduce in a rather natural way the special results by earlier workers.

The true self-avoiding walk is generalised by admitting that the self-avoidance parameter depends on the past history of the walk. For the generalised model the end-to-end distance exponent nu is calculated by a method that gives the same result as a Flory-type theory for the usual true self-avoiding walk. The exponent nu is predicted to be continuously variable with one of the parameters of the generalised model. The resulting expression for the exponent nu is confirmed by Monte Carlo simulations.

The dynamic critical exponent z for a system of nearest-neighbour interacting Heisenberg spins on a two-dimensional Sierpinski-type fractal is obtained by applying a real space renormalisation group transformation of the equations of motion.

Fluctuation arguments combined with basic features of the geometry of self-avoiding walks (SAWs) are shown to lead to zero transition temperature T_c for the Ising model on a SAW. The Ising model on a non-random fractal representing the essential geometrical features of SAWs is treated exactly by a decimation method. It also leads to T_c=0, and to Ising SAW critical behavior different from that on an Ising chain.

Random walks on self-avoiding walks (SAWs) are studied using Monte Carlo techniques on a square lattice (with nearest-neighbour hopping along the chain and between SAW points which are nearest neighbours on the embedding lattice). The average of the square of the end-to-end distance for random walks of t steps on SAWs of length N is fitted to the scaling forms (R_t²) varies as N^delta t^k (for t<<N^theta ) and (R_t²) varies as N^{2 nu s} (for t>or approximately=N^theta ), where theta approximately=2 nu s/k; nu s being the average end-to-end distance exponent for SAWs. The observed value of the exponent delta is supported by the authors' real space renormalisation group result for the conductivity of SAW chains. The exponent k has been related to the 'effective' fractal dimension of the SAW chain.

The authors show that at the disorder variety, the n-point correlation functions of the checkerboard Potts model have a simple causal structure, and decay as simple exponentials for space-like separations. An exact expression for the susceptibility in the Ising case is obtained. The result is additional evidence in favour of the conjectured S₄ symmetry of the checkerboard Potts model in the presence of a finite field.

A two-step renormalisation group approach-a Migdal-Kadanoff (or decimation) transformation (MK) followed by a mean-field renormalisation group (MFRG)-has been applied to random bond Ising models on d-dimensional hypercubic lattices. Calculations of the critical couplings and percolation concentrations for two and three dimensions show much improvement, even for small size clusters, compared with the results of separate MK or MFRG calculations. Critical exponents nu _T and nu _p are also improved in some cases. Estimates of critical curves in the concentration-temperature plane are also obtained for the two- and three-dimensional models.

A Flory-type calculation of the fractal dimensionality of the shortest path in a percolation cluster near the threshold is introduced. The result is roughly consistent with previous predictions or numerical simulations.

A computer simulation study is presented for directed growth in a three-dimensional kinetic growth model for irreversible gelation. Finite size scaling theory is used to extract critical exponents for the gel-sol transition. In contrast to percolation, the exponents for directed and undirected gelation appear to be the same.

The Aubry model near the critical region is analysed with the numerical renormalisation group method. The transition is shown to be energy dependent and the self-similarity is demonstrated.

The authors list all possible representations of semisimple groups G=G₁*G₂*. . .*G_k (k>or=3) for finite N=2 supersymmetric Yang-Mills theories, but without consideration of the cases of pseudoreal representations.

A Hamiltonian structure is constructed explicitly for a class of isomonodromy deformation equations for ordinary differential equations with one irregular singularity. The Hamiltonian turns out to be closely related to a differential form, which is known to be naturally associated to isomonodromy deformation equations. The monodromy data are constants of motion. The functional independence of the formal monodromy exponent and Stokes matrices is demonstrated by an example.

The author shows that certain classes of coupled Schrodinger equations in one dimension which appear in various physical applications can be uncoupled by the application of an appropriate Darboux transformation.

It is shown that the replica trick fails to give the correct non-perturbative result for the two-point function S₂ of the Gaussian unitary ensemble of N*N random matrices. The failure arises from an incorrect description of the symmetries of the random-matrix system in the limit N to infinity . The correct description, which involves integration over both non-compact and compact degrees of freedom, is obtained by using the method of superfields. Some implications for the localisation transition in disordered electronic systems and the theory of the quantised Hall effect are suggested.

Solutions depending on the noise scale epsilon in the same way as the Gibbs distribution are typical for systems with detailed balance (uniform in epsilon ), but apply also more generally. The author gives criteria for their existence as well as conditions for an elementary evaluation.

The authors extend the microscopic droplet theory of Ising systems, developed recently, to incorporate the effects of an ordering field. The theory yields a free energy which is the solution of a renormalisation group equation describing droplet nesting, and which has a full scaling form. The behaviour near the coexistence curve is investigated and found to display the essential singularity suggested by primitive droplet models, but with parameters renormalised in a physically intelligible way. The droplet population function implied by the theory is parametrised in a simple way, in space dimension d=2, to yield predictions for various thermodynamic properties, including the equation of state, which are found to be in fair accord with series-based results. The structure of the field-dependent droplet number distributions is investigated and contrasted with the forms assumed in phenomenological droplet theories. The domain of validity of the theory is assessed. It is concluded that the theory fails to describe (sufficiently) large droplets in the presence of a finite field, and is thus in principle trustworthy only in a region close to the coexistence curve.

Goldstone mode singularities in non-ordering correlation functions of the O(N)-symmetric phi ⁴ theory are investigated in 4- epsilon dimensions. Functions constructed from non-invariant combinations of the transverse operator pi ² and the longitudinal operators sigma ² and sigma generally exhibit severe infrared divergences on the coexistence curve. In low-order perturbative calculations of, for example, the specific heat, however, these divergences cancel. Such cancellations are shown to be a general feature of O(N)-symmetric correlation functions. Non-leading singularities remain, and lead to momentum dependence which can be expressed as a double power series in mod p mod ^epsilon and mod p mod ^{2- epsilon} .

Lehr considers that more details are required about the hidden variable model proposed by Liddy (1983, 1984) before it can be said to be a complete objective local hidden variable theory.

Liddy in his reply to Lehr's comment, while not wholly agreeing with the criticism, does concede that the theories he presented are not totally viable.

We would like to point out that the theoretical results published in our recent paper 'Quantum nucleation in the physical kinetics of phase transitions' were derived in essentially equivalent form, if perhaps less economically, by I M Lifshitz and Yu Kagan (1972 Sov. Phys.-JETP35 206). We are indebted to Dr Dietrich Stauffer of the University of Cologne for bringing this earlier work of Lifshitz and Kagan to our attention.