Table of contents

The authors obtain exact expressions for the (1+1)-dimensional Q ball solutions for a wide class of SO(2)-invariant potentials of the form 1/2 mu ² phi ²-1/6g² phi ⁿ⁺²+1/12 lambda ² phi ²ⁿ⁺² where n=1, 2, 3,. . .

In a recent paper De, Konar and Ray (see ibid., vol. 19, p.3693, (1986)), obtained the general embedding SO(4) solutions to the equation which the authors derived for the general spherically symmetric gauge potentials in the SU(N) sourceless Yang-Mills theory. They discuss their physical meaning and point out that the only analytic solutions are the instanton ones.

The author expresses the partition functions of the ADE lattice models on the torus in terms of partition function of six-vertex models and which can be computed in the continuum limit. Thus the author recovers the partition function of minimal models with C<1 and new ones with C=1.

The author shows that half-range completeness and half-range orthogonality relations hold for the stationary one-dimensional Fokker-Planck transport problem for some velocity-dependent external forces and general boundary conditions.

The authors find exact solutions of the integral form and the corresponding eigenvalues for the non-polynomial oscillator V(x)=x²+λx²/(1+gx²). The solutions are shown to exist when the parameter λ is constrained to satisfy a relation (of supersymmetric origin) involving the coupling constant g.

The authors have extended the formalism of Vinogradov and Krasil'shchik (1984) for the coupled system of Benny-Kaup equations. It is explicitly demonstrated that it is possible to generate new non-local symmetries for this coupled equation if they consider D^-1u= integral u dx to be also in the space of independent variables.

A Hamiltonian formulation of the particle representation of solitons of extended hierarchies of evolution equations is presented. The extension is related to the addition of new terms with x-dependent coefficients.

The authors examine the level statistics of Harper's equation, a simple model of a one-dimensional quasiperiodic system, through the distribution of the spacings between adjacent eigenvalues, each normalised by the average local density of states. They present exact results for the distribution for both small and large amplitude alpha of the incommensurate potential, showing it has a simple form in each of these limits. They show numerically that these distributions are preserved as alpha approaches its critical value 2 from each side, and also that at the critical point the distribution of the normalised spacings has a distinct but simple form.

The authors propose a generalisation of Parisi's replica symmetry breaking scheme (1979) to spin glasses (or optimisation problems) described by a set of order parameters Q^(r)_{alpha 1. . . alpha r}, or equivalently by a global order parameter G( sigma _alpha ), rather than by Q^(r)_{alpha 1 alpha 2} alone. They study the particular case of Derrida's p-spin model (1981) in the very dilute limit. The model is solved exactly in the large p limit and a freezing transition occurs. Using replicas and a single step of their replica symmetry breaking ansatz they recover the exact result. They discuss the implications for the global order parameter G( sigma _alpha ) when more than one step in the replica symmetry breaking process is taken.

Temperley's model (1952) of self-supporting stackings of circles in a triangular lattice array against a line wall is solved exactly in terms of q hypergeometric functions. For N circles, the number of different configurations is described by the large-N asymptotic law A lambda ^N, with A=0.312 36. . . and lambda =1.735 66. . . .

The authors present an efficient algorithm to determine the backbone (the current-carrying part of a lattice) in the random resistor network problem. It reduces mainly to the determination of the cluster structure on two lattices. Its efficiency is due to the use of the dual lattice and, therefore, is specific to the two-dimensional case.

Charge transport in a stirred percolating system of a conductor-insulator mixture (a model for the microemulsion) is studied by computer simulation in two dimensions. The flow of charge from source to sink via nearest-neighbour charge exchange on the transient network of the conducting particles is considered. For the particle jump accompanied by a local charge transfer, the conductivity depends non-monotonically on the concentration p of the conducting particles and exhibits a maximum near the percolation threshold. The maximum vanishes and the conductivity rises with concentration p with a non-universal power on increasing the frequency of charge exchange.

Hydrodynamic dispersion in random network models of porous media near the percolation threshold is investigated. This is done by studying the random walk of a particle in flow through the random network. Various scaling regimes (which depend on the Peclet number which is the ratio of diffusion and convection times) are identified, and the scaling relations for the mean-square displacement of the walk, both in the direction of macroscopic flow and in directions perpendicular to the macroscopic flow, are derived and related to those of anomalous diffusion on percolation clusters. It is shown that dispersion can give rise to superdiffusion in which the mean-square displacements of the random walk grow with time faster than linearly, while the spectral dimension of such random walks can be significantly larger than two, which is the critical dimension for diffusion on fractal systems. The author proposes a new equation for the probability density of finding the random walker at a point at a given time and discusses a method by which the probability density for first passage times of the walker can be determined.

In a bounded-synapses version of Hopfield's model (1984) for neural networks the quasienergy of a given memory, which is approximately equal to the depth of the corresponding energy well is calculated exactly by treating the change of a synaptic strength on learning as a random walk within bounds. Attractors corresponding to stored memories are found to be considerably flattened before serious retrieval errors arise. This allows dream sleep to be interpreted as random recall and relearning of fresh strong memories, in order to stack them on top of weak incidental memory imprints of a day.

A rule of synaptic modification in neural networks is proposed under a principle which minimises 'free energy' with synaptic strength roughly bounded. The rule does not allow the overloading of memories which can be a cause of failure in the memory process and it stabilises the synaptic connection, provided that memories are properly stored.

A quasiperiodic pattern (or quasicrystal) is constructed in real four-dimensional Euclidean space, having the non-crystallographic reflection group (3, 3, 5) of order 14 400 as its point group. It is obtained as a projection of the eight-dimensional lattice E₈, and has as a cross section a three-dimensional quasicrystal with icosahedral symmetry.

The possible non-crystallographic point group of the decagonal quasicrystal phase of Al-Mn alloys has been shown by Bendersky (1986) to be either D_10h or C_10h. For the physically irreducible representations of these groups, the authors derive the Clebsch-Gordan products extended integrity bases, stability spaces and tensorial covariants. The point groups which can arise in phase transitions are determined along with corresponding tensorial parameters which could drive the transition. It is shown that equilibrium tensorial properties whose components transform as the components of the electrogyration or elasto-optic tensors can distinguish between the D_10h and C_10h point group symmetry of the decagonal phase.

Two recursion formulae are deduced for the construction of inner- and outer-product isoscalar factors, respectively, for the symmetric group, by which the corresponding coupling coefficients can be easily obtained. A formula for the evaluation of the isoscalar factor for the subgroup chain U(N) contains/implies U(N-1) is proposed based on the relation between the outer-product coupling coefficient for the standard basis of the symmetric group and Clebsch-Gordan coefficient for the canonical basis of the unitary group. The calculation procedures are presented and the properties of those coefficients are investigated.

A method for calculating the eigenvectors of a ring system is developed. The system can be one chain or many coupled chains. A sample calculation demonstrates that this method is accurate, efficient and can be used to handle large systems.

The authors present a systematic method of classifying and constructing invariants for Lagrangians containing arbitrary polynomial non-linear potentials. It is based on the assumption that these Lagrangians are invariant under contact groups of transformations. For a finite number of degrees of freedom they can prove integrability for a large class of polynomial potentials. The method can be extended in several directions.

The Birkhoff-Moser theorem guarantees the existence of an analytical reduction to its normal form of an area-preserving map in the neighbourhood of an unstable fixed point. The stable and unstable manifolds emanating from this point are then simply the images of the axes in the normal coordinates. For sufficiently high parameter values of the standard map, the series for the manifolds can be resummed into a closed form, providing good approximations to the first loops of the manifolds. The authors thus obtain simple approximations for a pair of homoclinic orbits for parameters where most orbits are chaotic. The Frenkel-Kontorova model consists of an infinite sequence of equal springs and masses under the action of a periodic potential. The configuration where all the masses lie at the minima of the potential corresponds to the unstable fixed point of the standard map. A soliton or discommensuration, where one of the minima is missed, is represented by a homoclinic orbit. Using the normal form they calculate the soliton energy and its pinning energy. Good agreement is found with the pinning energy of Pokrovsky (1981) obtained as a perturbation from the continuum approximation.

Using the Hirota formalism, Gibbon et al (1976), have shown that the evolution equation u_t+u_x-u_xxt+(4u²+2y_xz_t)_x=0 with u=y_t=z_x, has the same solitary wave as the regularised long wave (RLW) equation u_t+u_x-u_xxt+6(u²)_x=0 and an exact two-soliton solution describing the elastic collision of two sech² profile solitary waves. Performing a more detailed analysis the authors show that the two-soliton solution can also represent other processes like the resonant or the singular collision of two RLW-type solitary waves. The interaction type depends on the values of a characteristic parameter of the solution. They also prove that with the bilinear form associated with the evolution equation, a three-soliton solution of the Hirota type cannot exist. They then study the equation u_t+u_x-u_xxt+3(u²)_x+6u_tz_x=0 with u=z_t, associated with another bilinear form, which has the same solitary wave as the evolution equation. They prove the existence of N-soliton solutions, for arbitrary N, and analyse the behaviour of the solitonic solutions. As in the first case, the two-soliton solution can describe elastic, resonant or singular interaction of two RLW-type solitary waves. A remarkable feature of the resonant triad is that it always involves one positive and two negative waves. This triad corresponds to a fundamental vertex for the analysis of the elastic soliton-antisoliton interaction.

It is shown that the system of non-linear differential equations y_k=y_k(y_k+1-y_k-1) (k=1, 2, . . . .; y₀=0) can be reduced to the system of linear differential equations delta _k=4 delta _k+1(k=1.2. . .).

The authors consider the equation of motion for a complex classical field in a (3+1)-dimensional ϕ⁶ model. The resultant complex non-linear Klein-Gordon equation is solved using an ansatz in which the envelope satisfies a scalar non-linear Klein-Gordon equation and the carrier satisfies either a wave equation or Laplace's equation with additional constraints. They use the results of the symmetry reduction method to exactly solve both the carrier equations and the envelope equation. The latter has recently been analysed and here they only briefly discuss the types of the appropriate solutions. However, they present a detailed discussion on one particular solution which is physically important. It bifurcates both in real space and in phase space. Possible physical applications have been outlined in the last section and they include superfluidity, superconductivity, liquid crystals and helicoidal metamagnets.

The derivation of various theorems dealing with the equivalence of electric and magnetic sources and the duality of the radiated fields in isotropic chiral media (D= in E+ beta in Del D*E, B= mu H+ beta mu Del *H) is given.

The Jaynes-Cummings model with multiphoton transitions in a cavity is examined, and an exact solution of the master equation for the density matrix is found. Absorption and emission spectra are investigated.

The path integral treatment of the hydrogen atom in a spherical space is discussed. The dynamical group SU(1,1) of the system is used for path integration. By mapping the radial path integral onto the SU(1,1) manifold, the energy spectrum and the normalised wavefunctions are obtained. In the flat space limit, the standard hydrogen spectrum and the corresponding normalised energy eigenfunctions are recovered. The scattering states are also found in the limit.

The author is concerned with discussing a matrix diagonalisation problem which occurs when one attempts to estimate the ground-state energy of a Bose fluid. The matrix problem comes about when one makes Bogoliubov's approximation to the Hamiltonian of an interacting system. The Hamiltonian is then quadratic in creation and annihilation operators. The ground-state energy can be computed if one solves an appropriate matrix diagonalisation problem. Here the roots of the characteristic polynomial for the matrix problem turn out to be the turning points of the roots of some simpler polynomials which depend on a parameter. Using this fact one can estimate them.

Using the radial ladder operators A⁺_l and A^-_l defined in solving the hydrogen atom radial equation with the factorisation method, the author develops a new device by which the calculations on the average value of r^s, the inner product of the radial wavefunctions and the matrix element of r are simplified and by which some new recurrence relations are derived.

The convergence radii of the Rayleigh-Schrodinger perturbation series for some bounded oscillators are calculated. The eigenvalues with quantum numbers n=0 and n=2, as well as those with n=1 and n=3, are found to cross at a pair of conjugate branch points of order two in the complex plane. Higher excited states are briefly discussed.

Formulae are derived for the average value and for the dispersion of the spectrum of an electronic Hamiltonian in a finite-dimensional, antisymmetric and spin-adapted space. Both the moments are expressed in terms of the corresponding two-electron quantities. The derivation is based on the theory of spin-adapted reduced Hamiltonians.

The statistical properties of energy spectra of a classical chaotic system are investigated, in particular for the effect of broken symmetry. Small symmetry breaking terms produce significant deviations from a Wigner distribution and also characteristic differences to the Delta ₃ function of the Gaussian orthogonal ensemble. Furthermore, a saturation effect for Delta ₃ was found as predicted in the literature.

The Schrodinger equation for the E⊗ε Jahn-Teller and Rabi systems in Bargmann's Hilbert space (1962) is a system of two ordinary differential equations of first order for the spin up and down components of the wavefunctions. This system has two regular and one irregular singular points. The energy eigenvalues are selected by the requirement that the solutions belong to the space of entire functions. The differential equations of the generalised spheroidal wavefunctions have the same singular points and the same exponents at each singular point. It is therefore conjectured that the component wavefunctions in the excited state i can be expanded in i+4 generalised spheroidal wavefunctions. The energy eigenvalues ν⁽ⁱ⁾ (i=0,1, . . ., 5) calculated with the conjectured component wavefunctions agree with numerical values within the computational error. The same is true for the coefficients of the Neumann expansion of the component wavefunctions. A proof is still missing.

By means of a physically interesting example, it is shown how the mathematical theory of quantum stochastic calculus can be used in modelling physical systems. The example is the so-called electron shelving effect. The fluorescent light emitted by a single atom or ion presents periods of darkness if two transitions, one weak and one strong, are simultaneously driven. By a previously developed theory of counting processes, based on quantum stochastic differential equations, the full statistics of the emitted photons is obtained. In particular, the results on the duration of bright and dark periods, previously obtained by Cohen-Tannoudji and Dalibard (1986), are completely confirmed.

The scaling hypothesis on the 'singular' part of the free-energy density of a finite system is examined in the context of a relativistic Bose gas confined to an Einstein universe of radius R. Finite-size effects in the various thermodynamic properties of the system, such as the free energy, the specific heat, the isothermal compressibility and the condensate density, are predicted in the regions of both first-order (T<T_c) and second-order (T approximately=T_c) phase transitions. To test these predictions, a detailed analytical study is carried out which includes the possibility of particle-antiparticle pair production in the system. The various predictions of the scaling hypothesis are fully borne out and the scaling functions governing the critical behaviour of the system are found to be universal, irrespective of the severity of the relativistic effects.

The authors investigate the statistical properties of local variables, coarse grained in time, in systems undergoing continuous phase transitions. A general theory is developed which suggests universal scaling behaviour in the limit of large coarse-graining times. The predictions are substantiated with Monte Carlo studies of two-dimensional scalar models. A simple fractal model of the time profile of the ordering variable is developed which captures and illuminates its timescale-invariant character. The theory is used to study the critical behaviour of local resonance lineshapes. In the two-dimensional case studied explicitly the critical slowing down is shown to drive a crossover to a split slow-motion lineshape.

The authors calculate the width of the growing interface of ballistic aggregation in the limit in which the range of the sticking interaction between the particles becomes infinite. They derive a scaling form for the width, and they compute the short- and long-time exponents finding nu =3/4 and alpha =1/2. Furthermore, they find that the crossover exponent defining the argument of the scaling function is gamma =1/2. They compare these exact results with computer simulations, finding excellent agreement. They also discuss the relation of these results to those of ordinary finite-range ballistic aggregation. Finally, they present a simple expression for the density of all ballistic aggregation clusters, regardless of the range of the interaction, which agrees with known results and interpolates between the infinite- and finite-range cases.

Eigenspectra of the critical quantum Ashkin-Teller and Potts chains with free boundaries can be obtained from that of the XXZ chain with free boundaries and a complex surface field. By deriving and solving numerically the Bethe ansatz equations for such boundaries the authors obtain eigenenergies of XXZ chains of up to 512 sites. The conformal anomaly and surface exponents of the quantum XXZ, Ashkin-Teller, and Potts chains are calculated by exploiting their relations with the mass gap amplitudes as predicted by conformal invariance.

A stochastic equation for lattice theories is written, which describes a Markov process on a space lattice evolving in (stochastic) time. At the cost of requiring the construction of the drift function this reduces one dimension in numerical simulations, as compared to Monte Carlo methods. The drift can be obtained either from the asymptotic solution of an auxiliary equation or from a ground state ansatz. It is shown that for Abelian theories a drift can be constructed from ground state ansatze which are exact eigenstates of Hamiltonians with the same continuum limit as the Kogut-Susskind Hamiltonian. Lattice observables may be obtained from stochastic time correlations. In addition, a new method is obtained to measure the lowest excited state (mass gap) from the exit times of the stochastic process from a bounded region. In some cases the mass gap may be obtained at weak coupling from the theory of small random perturbations of dynamical systems.

To calculate thermodynamic functions in statistical mechanics the authors propose a new variational approach which has as particular cases the approaches of Bogoliubov, Oguchi and Tsallis-da Silva (1982), but introduces some improvements, mainly in the short-range order effects above the critical temperature. It requires a choice of a trial Hamiltonian that defines an approximate ensemble through which the thermodynamic functions may be calculated. To check the efficiency of their approach they apply it to the ferromagnetic Ising model by using the simplest possible trial Hamiltonian.

For pt.I., see ibid., vol.18, P.3181(1985). The theta -point (collapse transition) is examined using a series analysis method for self-avoiding walks on the tetrahedral and square lattices. The results are, as a whole, compatible with Moore's conjecture (1977) that the order of transition is first in two dimensions while second in three dimensions. The temperature dependence of an exponent delta for the end-distance distribution is estimated together with those of exponents nu and gamma . In particular, the author has obtained that delta is 2.22+or-0.05 at the theta -point in two dimensions and 2.06+or-0.04 in three dimensions.

The coherent medium approximation of Odagaki and Lax (1981) is generalised to the trapping model. The frequency-dependent diffusion constant in the d-dimensional hypercubic lattice is studied when the jump rate obeys a bimodal distribution. The coherent medium approximation gives the correct static diffusion constant. The imaginary part of the AC part of the diffusion constant vanishes linearly in frequency omega when d>2, as omega ln omega when d=2 and as omega ^d/2 when d<2. The corresponding real part vanishes quadratically in frequency when d>4, as omega ² ln omega when d=4 and as omega ^d/2 when d<4. The termite limit is studied by taking the limit that one of the two jump rates (probability p) becomes infinite. In the termite diffusion the static diffusion constant is critical at p=1 and the critical exponents are the same as those for the termite diffusion in the hopping model. The ant (or ant lion) limit is defined by the limit of one jump rate being zero. The imaginary and real parts of the AC diffusion constant vanish linearly and quadratically in frequency, respectively. The critical exponent of the leading real part of the diffusion constant is one less than the corresponding exponent for the ant diffusion in the hopping model below the percolation threshold, while the leading imaginary part has the same critical exponent as those in the hopping model.

The commuting diagonal-to-diagonal transfer matrices of the self-dual Potts and Ashkin-Teller models on the square lattice are shown to satisfy special functional equations called inversion identities. These identities generalise the known local inversion or unitarity relations for interaction-round-a-face or IRF models satisfying Yang-Baxter or star-triangle equations.

The commuting row-to-row transfer matrices of magnetic hard squares on the multicritical T manifold are shown to satisfy a special functional equation called an inversion identity.For strip widths up to N=32, these equations are solved numerically for the transfer matrix eigenvalues. The central charge or conformal anomaly c=1 is obtained from 1/N² corrections to the exact bulk free energy and scaling dimensions are obtained from the eigenvalue gaps. The magnetic scaling dimension is x_m=1/8 and the sublattice density difference scaling dimension is x_e=1/9(2-y), where y=2 lambda / pi and the interaction-dependent crossing parameter lambda varies between 0 and 2 pi /3. Scaling dimensions for further operators fall in the sequence x_n=n²x_e where n=1,2,3,4. The critical exponents also vary continuously along the multicritical line and are given by alpha =(14-9y)/(16-9y), beta _m=9(2-y)/16(16-9y), beta _e=1/2(16-9y), nu =9(2-y)/2(16-9y) and delta =15. These exponents are simply related to the exponents of the critical eight-vertex and Ashkin-Teller models, which also exhibit Z₂*Z₂ symmetry, and the authors argue that these models lie in the same universality class.

Certain classes of correlated site-percolation problems (or correlated spreading phenomena) on Bethe lattices are analysed exactly. The author's analysis of percolation of, e.g., occupied sites, requires that spreading of clusters of occupied sites is determined by a finite number of conditional probabilities. A condition specifying the percolation threshold is provided, as well as expressions for the percolation probability and average cluster size. Previous results for random and nearest-neighbour Ising-model distributions are recovered as special cases. Results are illustrated with examples for equilibrium and non-equilibrium distributions, the latter obtained via irreversible cooperative filling. The author also discuss 'two-phase percolation' for distributions with no occupied NN pairs of sites, correlated bond percolation and other problems.

The asymmetry and prolateness of site percolation clusters and lattice animals are determined for two and three dimensions, with agreement with in expansion results.

This author proves that the maximisation of the entropy in non-equilibrium is equivalent to the exploitation of the entropy inequality in extended thermodynamics.

The authors obtain, from a thermodynamic basis, evolution equations for the kinetic and the potential contributions to the viscous pressure tensor for a real gas. The macroscopic predictions of the theory, previously checked from kinetic theory for ideal gases up to first order in the density, are also confirmed for real gases, up to second order in the density.

The kinetic equation for a copolymerisation system is established following Flory's two assumptions. The authors solve the kinetic equation for the A_f+A₂ system with arbitrary initial conditions. The exact solution for a copolymerisation system with constant deposition is also obtained. The post-gel solutions corresponding to Stockmayer's (1944) and Flory's (1936) models are investigated. The pre-gel solution to the kinetic equation for the A_f+A_g system with arbitrary initial conditions is given.

A perturbation scheme in non-equilibrium thermo-field dynamics is developed by taking into account the initial-state correlations. Two methods are proposed: in the first method the time dependence of amplitudes and their initial values are evaluated by two independent perturbation schemes while in the second method the two perturbation schemes are combined by use of a fictitious past.

The performance of neural network models with arbitrary non-linearity and Gaussian external noise superimposed on the synaptic efficacies is analysed. The memory function, though surprisingly robust, gradually fades out as the noise level is increased. In the low-noise limit the best performance is at zero temperature. There is a noise range, however, where optimal performance is obtained at a non-zero temperature.

The principles of spin gauge theories are explained. A particular spin gauge symmetry within the Clifford algebra C_2.6 is shown to give the correct GSW electroweak interactions for the electron-neutrino system. A new concept of mass is introduced, the electron mass being interpreted as an interaction with the 'frame field', which is proportional to the spacetime dependent Dirac matrices ( gamma ^u( chi )). Including the frame field in the 'extended covariant derivative' Delta ^mu and calculating ( Delta ^mu , Delta ^v) gives, along with the boson Lagrangian kinetic terms, exactly the correct photon, W and Z mass matrix. Transformation of the lepton extended covariant derivative to the 'quark representation' of the Clifford algebra, which is determined by the electromagnetic coupling constants, reproduces the GSW interactions for the up and down quarks. Thus, for the first generation electroweak theory, the 'Higgs-Kibble mechanism' is replaced by the frame field concept of mass. The models studied indicate that an energy associated with the frame field is approximately three times the W boson rest mass M_W. A refinement of the theory suggests a fermion mass of the order of M_W.

The Rarita-Schwinger field coupled to an external electromagnetic field is reduced to a constrained mechanical model. It is shown that, guided by the mechanical model with the linear supplementary condition, both the non-positive definiteness of the anticommutators and the non-causal modes of propagation (the Rarita-Schwinger paradoxes) have the same origin.

The group-theoretical side of N-pseudoreductions is discussed. The resulting equations are shown to be easy transformations of the N-KdV hierarchy.

It is shown that the gap separating the positive and negative energy states of the Dirac electron in the inhomogeneous, time-independent magnetic field does not shrink when the field gets stronger.

It is shown that exact solutions of the doubly anharmonic oscillator in the form of integrals can exist if the oscillator is given a supersymmetric form and the coupling constants satisfy a supersymmetric constraint.

A renormalisation group method is presented to derive the fractal dimension for the diffusion-limited aggregation (DLA). In a non-equilibrium steady state the dissipation energy and the conductivity of the surface layer play the roles of the free energy and the coupling constant in an equilibrium state. The renormalisation group transformation of the conductivity of the surface layer is obtained in the replacement of the coupling constant. The renormalisation group equation has a non-trivial solution where the derivative at the fixed point has a positive value of less than one. This is consistent with the theorem of minimum entropy production. The fractal dimension is expressed in terms of the derivative of the renormalisation group equation at the fixed point. The fractal dimension d_f=1.628 is found by using the 2*2 small cell renormalisation.

A new expression for the lacunarity of Sierpinski carpets is proposed. It turns out that the universality can be described with the new expression, the fractal dimension and the connectivity.

The author shows that the q-state bond-correlated percolation model (QBCPM), which is the percolation representation of the q-state Potts model (QPM), on the lattice without closed loops is equivalent to the bond random percolation model (BRPM) on the same lattice. Using such results and exact results for the BRPM on the linear and Bethe lattices, the author obtains exact cluster size distributions and the mean cluster size S for the QBCPM on the linear and Bethe lattices. The mean cluster sizes obtained from this method are the same as those obtained by more tedious exact calculations. Near the critical point, the average number of m site clusters per site, n_m, for the QBCPM on the linear and Bethe lattices may be written in the scaling form for large values of m, which is the geometrical basis for the scaling laws of critical exponents.

Percolation in a three-dimensional lattice, bounded by two free plane surfaces meeting at an angle chi , is studied within a real space renormalisation scheme. The edge critical behaviour at various surface and bulk transitions is analysed. The author does not find evidence for a first-order edge transition at higher values of alpha , as has recently been observed for an Ising model at an edge. Dependence of the edge scaling power y_e on the opening angle alpha has been confirmed.

Simulations for dispersion of diffusion at the percolation threshold of triangular and Bethe lattices show scaling behaviour. With 'topological' bias the authors find a maximum of the arrival time distribution at short times, a power-law decay for intermediate times and an exponential decay for long times.

Formulae for the amplitudes of each diagonal element in terms of the phases of the off-diagonal elements of the 3*3 unitary symmetric S matrix are given. These formulae are a great improvement and simplification on those given recently and have a form similar to the inverse relations. There are no problems with the choice of signs of square roots at any point within the Waldenstrom pyramid.