Table of contents

An analytic representation for the Bloch-Gruneisen integral, describing the electron-photon interaction contribution to the electrical resistivity of metals, is presented. The representation takes the form of an originally infinite series, truncated to K terms. The approximation is highly accurate yielding relative errors less than 1% for even a single term (K=1) for T<or= theta /4 and less than 0.1% with K=20 for all T<or=10 theta , where theta is the Debye temperature. The series approximation is shown to be superior to the previously suggested ninth-order polynomial in log( theta /T). Two other infinite series approximations are also discussed.

It is shown explicitly how a Bogoliubov type transformation acting on coherent states gives rise to states where the mean values of the field strengths keep the original values while the mean values of the squared strengths or the correlation functions can, within some bounds be prescribed independently.

The indefinitely growing self-avoiding walk is solved exactly on a regular fractal, the Sierpinski gasket. The fractal dimension D=In 5/In 2-1 differs from the value of D for the equilibrium self-avoiding walk on this fractal.

The authors investigate the behaviour of the maximum voltage drop across the bonds in a random resistor network above the percolation threshold. On the basis of numerical simulations on randomly diluted L*L square lattice networks, we find that the average value of the maximum voltage, (V_max(p;L)), exhibits a peak as a function of the bond concentration, p, which is located above the percolation threshold. This peak value appears to grow logarithmically with L, while the location of the peak appears to approach the percolation threshold very slowly as L increases. To help understand these results, they introduce the 'bubble' model, a quasi-one-dimensional structure in which the system length varies exponentially in the system width. This model is exactly soluble and a rather good description of percolation in systems of greater than one dimension can be obtained. Moreover, within the bubble model the peak value of the maximum voltage increases as In L, while the peak location approaches the percolation threshold as In(In L) In L, in good agreement with our numerical results.

The authors study the mass distribution of fractal objects growing by subsequent addition of units to the structure. It is demonstrated that in the limit of very large sizes such structures generally exhibit multifractal properties without defining a singular measure on the fractal support. Using the mass as a measure on the fractal, for a few simple deterministic processes they give the spectrum of corresponding fractal dimensions exactly. According to their results such geometrical multifractals should be very common and consist of parts which have a varying local fractal scaling of mass different from that of the whole object. They demonstrate the variation of multifractal properties as a function of the increasing size, in this way establishing for the calculation of multifractality an approach common in the studies of fractal properties of growing clusters.

A replica symmetric theory of the matching problem was given by Orland (1985) and by Mezard and Paris (1985). Here the authors investigate the stability of this solution. In the zero-temperature limit they find that the longitudinal spectrum is continuous and lies in the interval 0( lambda (2. They conclude that the solution of Mezard and Parisi is stable to longitudinal fluctuation. They give the explicit form of the longitudinal eigenvectors. The method used lends itself to dealing with replica symmetry broken solutions where necessary.

Directed animals on a square lattice with attractive nearest-neighbour and next-nearest-neighbour interactions are considered. Using an equivalence between the collapse problem and the directed site bond percolation problem, the bulk free energy per site is calculated exactly in the collapsed phase. It shows no singularity as the transition is approached from the collapsed side. For the three-dimensional directed animals, bulk free energy per site in the collapsed phase is calculated exactly if the interaction weights are related by an algebraic equation.

The authors compute numerically the asphericity of (i) percolation clusters at the percolation concentration on a square lattice, and (ii) Ising clusters at the critical temperature of an Ising model, also on a square lattice. These results are compared with various analytical calculations. For the percolation problem, they also compute the distribution of asphericities among different clusters. Finally, they show that the lattice itself does not give rise to any anisotropy.

The authors investigate the persistency of self-avoiding walks on the square lattice by studying the distribution of endpoint displacements projected in the direction of the first step. Exact enumeration and the constant-fugacity Monte Carlo data indicate that the mean displacement after N steps (x_N), increases logarithmically with N, while the higher odd moments (x^2k+1_N) vary as N^2kv In N, where v is the correlation length exponent. This unusual behaviour for the moments is found to arise from a simple scaling behaviour of the asymmetric component of the displacement distribution, which in the constant-fugacity ensemble can be written in the form A(x) approximately N^-2v In N exp(-bx/N^v), where N is the average number of steps in the walk at a fixed value of the fugacity and b is a coefficient of order unity.

The authors study diffusion in percolation systems at criticality in the presence of a constant bias field E. Using the exact enumeration method they show that the mean displacement of a random walker varies as (r(t)) approximately log t/A(E) where A/(E)=In((1+E)/(1-E)) for small E. More generally, diffusion on a given configuration is characterised by the probability P(r,t) that the random walker is on site r at time t. They find that the corresponding configurational average shows simple scaling behaviour and is described by a single exponent. In contrast their numerical results indicate that the averaged moments (P^q(t))= Sigma _rP^q(r,t)) are described by an infinite hierarchy of exponents. For zero bias field, however, all moments are determined by a single gap exponent.

The authors present a Monte Carlo algorithm based on 'logical transport' which determines the effective isotropic percolation and directed percolation thresholds of finite-size lattices. They apply the algorithm to bond and site directed percolation in two dimensions and determine the asymptotic percolation thresholds by finite-size scaling.

The authors study central-force percolation on a triangular lattice where the bonds are springs that can freely rotate around the nodes. All bonds in one preferred direction are infinitely rigid. In the other two directions a proportion p of the springs chosen at random are infinitely rigid while the rest have a finite strength. For this specific problem, the threshold at which the network becomes infinitely rigid corresponds to the usual bond-percolation threshold of a square lattice and is therefore exactly equal to 0.5. They use a transfer-matrix algorithm to study the elastic modulus of strips of width ranging from 2 to 32 and length 10⁵. They obtain an estimate of the critical exponent relative to the scaling of the elastic moduli with the strip width 0.99+or-0.03, close to s/v for the corresponding isotropic electric problem.

The authors present an accurate determination of the percolation transport exponent u=t/(s+t) in three dimensions, based on a transfer-matrix approach to the AC (frequency-dependent) conductivity, and a finite-size scaling analysis of the numerical data. The phase of the complex conductivity, the loss angle, assumes the universal value delta _c=( pi /2)(1-u) at low frequency at the percolation threshold. As a test of their numerical scheme, the two-dimensional exact duality result delta _c= pi /4 (i.e. s=t) is recovered with a very good accuracy. In three dimensions, their data are extrapolated to tan delta _c=0.54+or-0.03, i.e. u=0.69+or-0.02, whereas the usually accepted values s/v=0.85+or-0.04, t/v=2.2+or-0.1 yield u=0.72+or-0.02. This marginal disagreement can be attributed to ill behaved corrections to finite-size scaling.

The author shows that the conformal field theories describing two-dimensional critical behaviour with continuously varying exponents must have central charge c=1 if (a) there are no conserved spin-2 currents other than the stress tensor and (b) the marginal operator responsible for the line of fixed points does not mix with other operators. In such cases one may show the existence of a fixed line knowing only the four-point function of the marginal operator. The author applies this to the Ashkin-Teller model and the XY model with fourfold symmetry breaking.

The author investigates a possible way of constructing finite-dimensional indecomposable representations of the generalised Lie algebras. The spl(2,1) superalgebra has been studied as an example.

A new method of algebraisation of quantum mechanical eigenvalue equations is presented. In this method the dynamical algebra is represented on the space of group matrix elements. The ladder operators of the dynamical algebra are obtained from Infeld-Hull-Miller factorisations. The method is used to study the first Poschl-Teller equation even in the non-symmetric case. The energy spectrum and the exact normalised solutions are obtained in agreement with the results of non-algebraic methods.

The method of an earlier paper (see ibid., vol.20, p.4075 (1987)) is applied to the non-compact case to solve a family of second Poschl-Teller Morse-Rosen and Eckart equations with quantised coupling constants. Both discrete and continuous spectra, bound state and scattering wavefunctions (transmission coefficients) are found from the matrix elements of group representations,.

The 'accidental' degeneracy occurring in the quantum mechanical treatment of the ring-shaped potential V=- eta sigma ²r^-1+1/2q eta ² sigma ²(r sin theta )^-2 is explained by constructing an su(2) dynamical invariance algebra. The Schrodinger equation is solved in parabolic coordinates written in the framework of the Kustaanheimo-Stiefel transformation and the Hamilton-Jacobi equations are solved in ordinary parabolic coordinates. All finite trajectories are found to be periodic.

A study in the statistical mechanics of phase transitions has involved certain seemingly simple Feynman integrals which have yielded surprisingly significant mathematical results. The integrals enable the derivation of new transformation, summation and reduction formulae for some single and multiple variable hypergeometric series. The main result reported in this paper is a transformation formula, a special case of which is a summation formula for a symmetric combination of two non-terminating Clausen's hypergeometric series of unit argument.

For pt. I, see ibid., vol.20, p.4109 (1987). Further examples of the use of Feynman integrals enable the derivation of new properties of hypergeometric series including new analytic continuation formulae for a generalised hypergeometric series and for a Kampe de Feriet function. This motivates the derivation of two new summation formulae for a generalised hypergeometric series and furthermore leads to a natural generalisation of the H function. While the latter, as is well known, contains as particular cases most of the special functions of applied mathematics, it does not contain some of importance, for instance the Riemann zeta function nor indeed any polylogarithm. Our generalisation of the H function does contain the polylogarithm; it also contains the exact partition function of the Gaussian model from statistical mechanics. Another new result is the simple summation formula ₃F₂(1,1,3/2;2,2; x)=-(-)^-1 ln((1+(1-x)¹2/)/2).

The recursion method gives a good approximation of the density of states in the single-band case, where the recursion coefficients converge. When band gaps exist, it is known that they exhibit asymptotic undamped oscillations. In order to clarify these asymptotic behaviours, an analytic study of the recursion method applicable to the band-gap case is presented. The asymptotic properties of the system of orthogonal polynomials to which this method is closely related, are investigated by use of the concept of the exterior mapping function. Various relations, which connect the means of the recursion coefficients to the support of a spectrum, are obtained. The period of their asymptotic oscillations is written as a function of the support. The numerical results by Turchi, Ducastelle and Treglia (1982) can be well interpreted by the present theory. The asymptotic properties of the truncated basis are also examined.

Numerov's method and an extended version of it are introduced for computing eigenvalues of Schrodinger equations with potentials V(x) which are even functions with respect to x. Furthermore it is assumed that the wavefunctions tend to zero for x to +or- infinity . The derived results are compared with previously derived numerical data and with available exact values.

The authors investigate transient periodic orbits of dissipative invertible maps of R². Such orbits exist just before, in parameter space, a saddle-node pair is formed. They obtain numerically and analytically simple scaling laws for the duration of the transient, and for the region of initial conditions which evolve into transient periodic orbits. An estimate of this region is then obtained by the construction-after extension of the map to C²-of the stable manifolds of the two complex saddles in C² that bifurcate ino the real saddle-node pair.

The problem of finding ansatze for a non-linear Dirac equation which is invariant under the extended Poincare group is solved. With the help of these ansatze some multiparameter families of exact solutions of non-linear Dirac and Dirac-Maxwell equations are constructed.

It is shown that, when plane polarised light of frequency ω and polarisation angle ϕ₀ is scattered by harmonically bound electrons of natural frequency ω₀, the polarisation angle, ϕ, of the second harmonic (2ω) scattered light is given by tan phi =tan ϕ₀ sec θ if ω=ω₀, tan ϕ=0 if ω=1/2ω₀, tan ϕ=1/2 sin 2 ϕ₀/(cos²ϕ₀ cosθ-μ) if ω not= ω₀,ω not=1/2 ω₀ where μ=(ω ²₀-ω²)/(ω²₀ -4ω²) and θ is the angle of scattering. To an observer facing the incident light, ϕ₀ is the minimum angle that the electric vector makes with the plane of scattering; similarly ϕ is the minimum angle that the electric vector of the scattered light viewed by an observer facing it makes with that plane. The consequences are discussed.

This paper is the first of two in which the coupled cluster method (CCM) or exp(S) formalism is applied to two-component Fermi systems, the aim being to describe real metals and superconductivity. In this paper the authors concentrate on exact results and restrict ourselves to a ring approximation, applicable essentially to a high-density regime. They show that in the ground-state formalism the random phase approximation (RPA) can be formulated as a system of coupled, bilinear integral equations satisfied by functions associated with the so-called four-point functions of the system which provide a measure of the two-particle two-hole components in the true ground-state wavefunction. These equations are analysed in the dimensionless parameter formed by the ratio of the species masses and exact analytic solutions obtained. For Coulombic potentials (V₁₁V₂₂=V²₁₂) they show that the exact analytic solution is unique and obtain an expression for the correlation energy. For non-Coulombic potentials (V₁₁V₂₂ not=V²₁₂) they indicate how to obtain a possible analytic solution. An RPA-like treatment of the one- and two-body equations in the excited-state formalism is provided for completeness.

Using a fully coupled oscillator model for the quantum mechanical description of a harmonically bound electron in a radiation field, exact results about the dissipative properties of the system in thermal equilibrium are derived. In particular one gets, without any additional assumption, the classical equation of motion of an oscillating electron damped by radiation and the scattering cross section of the system for incident light.

Recent work on states bound by a central potential whose Laplacian is positive is extended to cover a wider class of potentials for which energy level ordering theorems can be proved. This class includes power law potentials in particular. Constraints on the moments of the radial distance are improved and similarly for the kinetic energy. These constraints are shown to provide tight bounds on the energies of ground states for power law potentials, and on their variation with angular momentum. They are also used to bound the position of the maximum of the wavefunction for these states.

A quantum Hamiltonian describing a set of interacting vortices is studied. A non-relativistic approximation for anomalous quantum electrodynamics in (2+1) dimensions is provided by this type of Hamiltonian. This model leads to fractional values of Hall conductance once the assumption that the external magnetic field pierces the plane creating the phenomenological vortices is granted. Conditions imposed on the wavefunction lead to a transverse conductivity which is quantised at least in the values sigma =(1/N)e²/h. N can be restricted to be an odd integer.

A formulation of supersymmetric quantum mechanics in terms of sesquilinear forms is presented. Otherwise occurring rather complicated domain questions are solved in a simple and natural way. The authors formulate the axioms and describe a number of models. Supersymmetry transformations are implemented as superunitary transformations in a space defined over a Grassmann algebra, generalised Bogoliubov transformations acting in that space are also formulated.

The supersymmetric massless states of a relativistic extension of Witten's supersymmetric quantum mechanics are shown to correspond to Abelian gauge fields realised by antisymmetric tensors. The massive states, which are only partially supersymmetric, include spin-1 and spin-0 particles. In particular the Maxwell and Proca fields are obtained from N=2 supersymmetric relativistic mechanics by first quantisation in much the same way as the Dirac theory arises in the N=1 case. The particles may be coupled supersymmetrically to external scalar and complex Hermitian tensor fields. The latter reduce in special cases to the Riemannian metric of external gravitation and the Kahler metric implied by a vector field coupling. All the couplings exhibit a quadrupole characteristic of the particles. In particular supersymmetry requires their electric charge and magnetic dipole moment to vanish, and there is no coupling to torsion in Riemann-Cartan spacetime. Quantisation in external fields yields covariant generalisations of the classical tensor field equations. Finally classical equations of translational and spin motion involving only real quantities are obtained from the Heisenberg equations.

The tunnelling of a single quantum spin is studied in the limit of large spin quantum number S. The problem is mapped onto a particle problem on the positive half-line with a Hamiltonian which is invariant under inversion x to 1/x. Not only the ground-state energy but also all the other energy levels and corresponding level splittings (if any) are computed by using the conventional WKB methods for the particle problem and an excellent agreement with numerical data is found.

Gives a short proof that any infinitely divisible even representation of a Clifford algebra or a CAR algebra is quasi-free, and of Hudson's central limit theorem (1973) for fermions. The author shows that the entropy of any even state of the CAR, Clifford or CCR algebra is less than or equal to the entropy of the quasi-free state with the same two-point function.

With the method of factorisation a new potential with free parameters is generated. This potential possesses exactly the same energy spectrum as the isotonic oscillator (harmonic oscillator with centripetal barrier). The wavefunctions and shift operators of the new system are also presented.

Bloch electrons in a magnetic field can be modelled by a one-dimensional Hamiltonian H=H(x, p), which is periodic in x and p. When beta , the ratio of Planck's constant h to the area of a unit cell is a rational number p/q, the spectrum is a set of q Bloch bands. If beta is perturbed away from this rational value, the spectrum becomes a Cantor set, but it is still possible to define a set of generalised Bloch waves which form a complete basis set for a given band. Projecting these states into the band, and taking matrix elements of the Hamiltonian leads to a new effective Hamiltonian H¹_v(x,p) and Planck constant h'_v describing the spectrum of the vth band. Since degrees of freedom have been eliminated by projecting into a band, this is a renormalisation group (RG) transformation. The RG transformation can be iterated indefinitely if the measure of the spectrum is zero or if beta is a Liouville number. The measure of the spectrum vanishes if H(x,p) has centres of threefold or fourfold symmetry in the phase plane, and the RG transformation explains the hierarchical structure of the spectrum observed in these cases.

The authors present a new technique to calculate ensemble averages of exponentials of quadratic boson operators, possibly multiplied by exponentials of linear boson operators. The method is based on the use of coherent states in the evaluation of matrix elements. Although this is a very simple basic idea, the technique enables the exact evaluation of several complicated averages from the theory of exciton transport. These averages, which could previously be given only approximately, are treated as examples to discuss the features of the method.

Presents the reduction to a normal form of a Grassmannian differential equation describing a classical fermionic system in the neighbourhood of an instability. As an application of the method presented the author shows that the anticommuting version of the normal form associated with (a) the Hopf bifurcation (a simple pair of imaginary eigenvalues) leads to the overdamped Grassmann harmonic oscillator, and (b) the resonant Hopf bifurcation 1: 1 (a double pair of semisimple imaginary eigenvalues) corresponds exactly to the Thirring model for Grassmann solitons.

Considers a supersymmetric formulation of the two-point function and shows explicitly that the underlying stochastic model symmetries can be implemented in the fermion-fermion sector through a compact parametrisation of the sigma _F matrix.

Considers random walks consisting of successive steps between neighbouring sites on a Cayley tree of arbitrary coordination number z= sigma +1. Each bond between neighbouring sites in assigned a conductance qg₀, where q is the number of times the bond is traversed in the random walk. The author presents an exact evaluation of the resistive susceptibility from which it is found that the end-to-end resistance averaged over all z^N walks of length N is given by (R)^(N) approximately N for sigma )1 and (R)^(N) approximately 1/2(1n N) for sigma =1, i.e. for a one-dimensional chain.

It is shown that within a two-parameter 3*3 cell to bond position space renormalisation group theory growing self-avoiding walks and growing self-avoiding trails on the square lattice are in the same universality class.

The authors present estimates for critical exponents of self-avoiding walks on a cubic lattice. They treat a grand canonical ensemble of walks with free ends in a Monte Carlo approach and make use of real space renormalisation ideas. Their estimate for v is 0.59+or-0.01. The estimate for gamma does not share the same degree of accuracy. However, they are able to pinpoint the source of this discrepancy. In addition, they define a quantity chi _L( beta ), the probability that a walk starting at the origin will end outside or at the border of a cube of side L. This quantity turns out to be quite suitable for a real space renormalisation group analysis.

The winding angle about a selected point of a random walk quantifies what is perhaps the simplest manifestation of entanglement a phenomenon of great importance in the study of polymers. The distribution of winding angles is studied here for ordinary or non-self-avoiding walks. New results highlight the crucial influence of the exclusion of a finite region about the selected point on the winding angle distribution. The results of this analysis are compared with simulations of random walks on a lattice, the agreement between the two is very good. There are, however, small and as yet unexplained discrepancies.

The Ising model on a two-dimensional quasi-crystal (the Penrose tiling) is studied. Using the correlation inequality and the duality transformation bounds for the critical temperature are obtained as 1.82<T_c<7. Monte Carlo simulations on finite lattices give an estimate of T_c=2.41+or-0.02. Finite size scaling analysis of the Monte Carlo data shows that the system belongs to the same universality class as the Ising model on the two-dimensional Bravais lattices. However, the finite size scaling forms do not reproduce the asymptotic limits in the range studied whereas, in the same range, the periodic lattices are known to behave as expected.

The authors study a modified Hopfield model of associative memory with a learning rule proposed by Personnaz et al (1985) for the special case of ultrametrically correlated patterns. The formula for the synaptic strength tells the 'teacher' how much stress to put on details compared to averages at each level. It is still a local rule if one assumes that the ultrametric correlation structure of the patterns is given a priori. The result is also given in terms of the Parisi function q(x). In a special limit they get the same result as Parga and Virasora (1986).

Chain-chain aggregation is studied by Monte Carlo simulations on the triangular lattice. N-conserving local deformations lead to snake-like motion or 'free-reptation' of the aggregating chains. The scaling of the chain-size distribution function is carefully examined and estimates of the static and dynamic exponents are obtained.

For pt. I see ibid, vol.19, p.L923 (1986). The RSOS-3 model is a sequence of two-dimensional solid-on-solid models with integer heights restricted between 1 and r-1 and nearest neighbours differing by 0, 1 or -1. It has an exact solution manifold with four regimes of distinct physical behaviour along which the height probabilities in the bulk of the system can be calculated. Using some results of the previous letter in this series, the authors calculate the height probabilities for two of the regimes. Appropriate order parameters are defined and their critical exponents calculated.

In order to characterise the interplay between quasiperiodicity and order in one dimension, the authors consider sequences of 0 and 1 generated by a circle map. These sequences, which generalise the Fibonacci sequence, describe the quasiperiodic ordering of atoms and vacancies on a line. They study in a quantitative way the unbounded fluctuation of the atomic positions WRT the average lattice. For some quadratic algebraic values of the rotation number the sequences can be generated by inflation rules, which proves their self-similarity. These rules, obtained by a renormalisation of the circle map generating the sequences, permit us, e.g. to explain the logarithmic divergence of the fluctuation. For some exceptional rotation numbers, the fluctuation diverges as N alpha , N being the system size. Whenever alpha >1/2, the quasiperiodic chain is therefore less 'rigid' than a random one.

A method to calculate perturbative corrections to the spectrum of a one-dimensional quantum chain in the scaling variable z is developed from the conformal invariance at its fixed point. Then, from the knowledge of certain correlation functions, one can calculate the first coefficients of the scaling functions. The method is applied successfully to the Ising model, where the corrections are also known from the exact solution. The first coefficients of the magnetic scaling functions of the Ising model are also obtained.

The density of partition function 'temperature' zeros for the two-dimensional spatially anisotropic Ising model in the absence of a magnetic field satisfies a linear homogeneous partial differential equation. A generalised form of this equation, for a non-zero specific heat exponent, can be derived by applying non-linear scaling to the expression for the specific heat as an integral over the density of zeros.

Classical ferromagnetic spin systems on self-similar fractal lattices are studied. By a variation of Simon's argument (1980), it is shown rigorously that the critical decay exponent Delta (which is characterised by ( phi _x phi _y)<or approximately=dist(x,y) at the critical point) satisfies the bound Delta <or=Q, where Q is the connectivity. The author also discusses some general features of the phase transitions and critical phenomena.

A general formula for the free energy of a polymeric mixture is given. Four terms are shown to be present the entropy of mixing the entropy of conformation, an isotropic interaction and an anisotropic interaction. The interaction parameters can be related to microscopic characteristics such as the degree of polymerisation (DP) and flexibility or persistence length. The DP dependence of the order-disorder transition of liquid crystal polymers is discussed.

Extended finite-size scaling and series expansion methods are used to determine the critical properties of the one-dimensional quantum Ising model with three-spin interaction. By supposing the presence of logarithmic corrections the universality class of the model is investigated. The results obtained by different methods support the conjecture that the model belongs to the same universality class as the four-state Potts model.

Indecomposable representations of the Lie algebra of the rotation group SO(3) are studied on the Heisenberg-Weyl basis, which is the basis for a subspace of a quotient space of the space of the universal enveloping algebra of the Heisenberg-Weyl algebra. The author obtains an infinite-dimensional indecomposable representation which induces the finite-dimensional representation on certain quotient spaces and subduces the infinite-dimensional indecomposable representations on invariant subspaces. The standard angular momentum representations of SO(3) are given as special cases.

A method for regularising various divergent sums is presented. The method involves the Mellin transform and is illustrated by examples.

Using linear stability analysis the authors argue that the asymptotic number of needles growing out of a center in a Laplacian process is two in all dimensions. Special arrangements in two dimensions can be solved by conformal mapping.

Comments that the Laplacian random walk with eta =1, recently introduced by Lyklema and Evertsz (1986), is the same process as the loop-erased self-avoiding random walk analysed previously by the author (1980). Some rigorous results about this process are reviewed.

The authors present a transfer matrix method for calculating the properties of directed compact site animals in two dimensions. The non-degeneracy of the largest eigenvalue of a truncated 50*50 matrix implies that the critical exponent theta =O. They also predict that the exponents v_{perpendicular to} =1/2 and v/sub ///=1.

Varying the concentration of cosurfactant in a microemulsion shifts the percolation threshold of the electrical conductivity but does not change the behaviour of the conductivity near the threshold as a function of temperature. The same effect is observed if one adds various alcohols.