rigid animals in three dimensions the number of rigid clusters c n is enumerated for the central force model on a site diluted fcc lattice the exponent vr governing the growth of the radius of gyration of the rigid animal is also calculated it is found that c n approximately n0 333n and vr 0 32 or 0 02

examination of the 1 d expansion method from exact enumeration for a self interacting self avoiding walk the 1 d expansion method for polymer chains is examined by comparing these expansions for several thermodynamic and structural quantities with the results of standard series analysis of exact enumeration data the comparisons cover a wide range of spatial dimensions d including non integer ones and are performed for particular values of interaction energy good agreement is generally found for d 4 whereas discrepancies become conspicuous as d decreases to d 2 reasonable values are obtained for the exponents v and gamma in d 2 4 by applying the coherent anomaly method of suzuki 1986 to our 1 d expansions through fifth order in d 1

heat fluctuation distribution from non equilibrium states a non equilibrium state is described by parameters alpha j aj which are ensemble averages of dynamical phase functions aj gamma by using a projection operator technique of zwanzig 1961 one can derive from the classical liouville equation an integro differential equation for g vj the amplitude for the probability that the aj have respective values vj these previously derived results are applied to the heat flux aj in a fluid and an approximate differential equation derived for g v j where j aj the first moment of this equation yields the catteneo vernotte equation 1958 with a set of auxiliary assumptions which are also employed in solving the equation for g the author obtains g g 0 1 g v j where g 0 kappa pi mu 3 2 exp mu v j 2 kappa is the einstein distribution and g is a sum of products of scalar combinations of j v j and del t and laguerre polynomials ln 12 z2 where z identical to mu kappa 1 2 v j the einstein distribution g 0 gives correctly nu 2 to o j2 when del t 0 but not in the presence of a temperature gradient

the projected area of the three dimensional self avoiding polygons in a plane calculates the exponent theta s governing the growth of the projected area of the three dimensional self avoiding polygons in a plane the series result suggests that theta s 2vsap where vsap is the correlation length exponent of the three dimensional self avoiding polygons

free energy of self interacting uniform stars we consider uniform star polymers on a variety of lattices with an additional contact interaction between pairs of vertices which are unit distance apart and are not joined by an edge of the star we present some rigorous results and other evidence which indicate that these systems have the same limiting free energy as self interacting self avoiding walks we discuss the extension of these results to other homeomorphism types and to systems with an additional surface interaction term

mapping between hopping on hierarchical structures and diffusion on a family of fractals the authors present a simple analytical approach to studying diffusion on hierarchical and fractal structures they show that hopping on one dimensional hierarchical structures can be mapped onto diffusion on a family of loopless fractals

diffusion on random hierarchical structures the authors study diffusion on a statistically self similar fractal substrate with random transition rates possessing a hierarchical structure the master equation is solved analytically by means of a recursion technique and different choices of random hierarchies for the transition rates are considered leading to algebraic stretched exponential or exponential logarithmic behaviour for the moments and the correlation decay making use of the central limit theorem analytical expressions for the fluctuation corrections to lowest order in the relative variances are derived for the first two cases where the behaviour is qualitatively the same as in the systems without fluctuations the sign of the corrections turns out to depend on the temporal and spatial scale at which fluctuations are externally suppressed they are negative if the diffusion is taking place at smaller scales positive otherwise the relevance to diffusion in a turbulent medium intermittency corrections is discussed

the fractal dimension of the minimum path in two and three dimensional percolation the authors calculate the fractal dimension d min of the shortest path l between two points on a percolation cluster where l approximately rd min and r is the pythagorean distance between the points they find d min 1 130 or 0 002 for d 2 and 1 34 or 0 01 for d 3

age distributions in physical systems recently vlad et al 1984 introduced a new stochastic description of memory effects based on a system of age dependent master equations adme in this paper the adme approach is applied to physical processes obeying a phenomenological master equation the joint probabilities and the correlation functions of the ages of different fluctuation states are computed the suggested method is extended to systems for which the transition rates are age dependent a physical interpretation of the main results is given by means of a system of age dependent integral equations the author proves that the adme formalism is a generalisation of the well known continuous time random walk theory ctrw

1 d expansions for the free energy of weakly embedded site animal models of branched polymers we give the 1 d expansion to order 1 d5 for the limiting reduced free energy of a weakly embedded two variable site animal model of branched polymers hence we are able to derive expansions for the free energy of related one variable models and the growth constant of weakly and strongly embedded trees again to order 1 d5 we argue that the free energy expansions are asymptotically correct for a small range of fugacities only exact results show that animals with a compact hypercubic structure make the overwhelming contribution to the total number of weakly embedded site animals especially for large d this result is not reflected in the free energy 1 d expansions where trees make the dominant contribution we also derive new exact enumeration data for lattice animals using intermediate calculations in the derivation of the 1 d expansions thus we give partition functions for the general d dimensional hypercubic lattice for one to 13 sites

generating functions and elementary young tableaux when constructing polynomial bases for group representations one is led naturally to consider a stretched product of states elementary multiplets are states that cannot be expressed as the stretched product of any other two states and they can be identified with terms occurring in certain generating functions in this paper the authors define an analogous stretched product of young tableaux together with elementary tableaux these may be used in a manner similar to elementary multiplets to construct generating functions they consider several examples where this method can be applied and in particular give new large n rank free branching rule generating functions for the group subgroup pairs su n su m contained in implied by su n m first four labels non zero so n contained in implied by su n first three labels non zero and sp 2n contained in implied by su 2n first five labels non zero this method of finding these generating functions uses the theory of grobner bases which allows the authors to make precise the notion of compatibility relations or forbidden products of elementary tableaux

rigid backbone a new geometry for percolation understood in both cases from the nodes linksblobs picture of percolation clusters in fact it is fair to say that in the vast majority of percolation problems encountered to date even continuum percolation 6 the basic geometric features of the percolation clusters such as the scaling of the number of singly connected bonds are identical and the differences arise only at the level of the dynamics for the class of models of interest here however geometric connection the existence of a bond does not imply complete elastic connection because the number of constraints associated with the existence of a bond is smaller than the number of degrees of freedom associated with the sites it connects the percolation threshold is higher8 than that of the corresponding standard percolation problem thorpe6 has suggested the name rigidity percolation for this class of models distinguishing it from the elasticity percolation universality class mentioned above in this paper we study the simplest rigidity percolation model namely the two dimensional central force model 8 ex

random walks on ultrametric spaces mean and variance of the range the authors study thermally activated hopping processes on regularly multifurcating ultrametric spaces ums and investigate the first two moments of the range rn where n is the number of steps paralleling the recent findings for cayley trees the knowledge of exact generating functions for these moments permits to draw conclusions about their behaviour for large n for the first moment the mean number of distinct sites visited sn they derive analytically the cross over from a linear behaviour in the transient regime gamma 1 to a sublinear increase sn approximately ngamma in the recurrent regime gamma 1 here as usual for ums the parameter gamma depends linearly on the temperature the second moment of rn the variance vn displays different behaviour corresponding to strongly transient gamma 2 not strongly transient 1 gamma 2 and recurrent gamma 1 random walks all results are verified by comparison to computer data obtained through simulations and through numerical series expansions

recursive sampling of random walks self avoiding walks in disordered media the author presents a new algorithm for simulating random walks which is simple versatile and efficient it uses recursive function calls and can be used to obtain unbiased samples with any given length distribution this makes it particularly useful in disordered geometries where the effective connectivity constant is not known a priori when applying it to self avoiding random walks on two dimensional media with quenched randomness he finds evidence for at least two new renormalization group fixed points

precise determination of the bond percolation thresholds and finite size scaling corrections for the sc fcc and bcc lattices extensive monte carlo simulations were performed to study bond percolation on the simple cubic s c face centered cubic f c c and body centered cubic b c c lattices using an epidemic kind of approach these simulations provide very precise values of the critical thresholds for each of the lattices pc s c 0 2488126 0 0000005 pc f c c 0 1201635 0 0000010 and pc b c c 0 1802875 0 0000010 for p close to pc the results follow the expected finite size and scaling behavior with values for the fisher exponent 2 189 0 002 the finite size correction exponent 0 64 0 02 and the scaling function exponent 0 445 0 01 confirmed to be universal

generalization of thermodynamic potentials including information in this work we analyze the connection between information theory and thermodynamics three different potentials are presented all related to the kullback leibler divergence these potentials represent three different conditions of energy and information exchange between a reservoir and a subsystem the first potential describes a situation where the two systems are exchanging energy without any external control the second potential considers the situation where the reservoir can get information through the state of another secondary system about the internal state of the subsystem this information can then be used by a maxwell demon to increase the free energy of the subsystem in this second scenario it is assumed that the energy cost of the information probe and the maxwell demon are at the expense of the reservoir finally a third case is considered where the information obtained through the secondary system is used not only to manipulate an internal maxwell demon but to perform directly work over the subsystem this last case involves a direct modification of the hamiltonian of the subsystem

derivation of exact product forms for the simple cubic lattice green function using fourier generating functions and lie group identities the simple cubic lattice green function is investigated where l1 l2 l3 is a set of integers w w1 iw2 is a complex variable and is a real parameter in the interval 0 in particular a new and direct method is used to prove that g 2n n n w and g n n n 1 w can be expressed in terms of a product of two 2f1 hypergeometric functions where n 0 1 2 in earlier work delves and joyce obtained these 2f1 product forms by solving complicated fourth order linear differential equations of the fuchsian type in this paper fourier generating functions and a known lie group addition theorem play crucial roles in the derivation of the product forms many term recurrence relations are also derived for g 2n n n w and g n n n w

diffusion limited aggregation near the percolation threshold abstract diffusion limited aggregation dla and dielectric breakdown db models have been used to simulate growth controlled by a laplacian field on a square lattice network a fraction near the percolation threshold c of the bonds had a high conductivity equal to 1 while the others had a low conductivity equal to r we used 10 5 r r 10 8 for db we find crossover from growth on an incipient percolation cluster with fractal dimensionality d 1 3 for small length scales to that on a uniform substrate d 1 7 for a large length scales the crossover length behaves as l r a with the crossover exponent a 0 25 the results were using a scaling theory

directed percolation front in a gradient the authors study numerically the structure of fronts generated in a directed percolation problem in two dimensions when the probability for a bond to be present decreases linearly along one direction they discuss several cases of relative orientation of the preferential direction with respect to the gradient they also consider the domany kinzel limit case of directed percolation and obtain analytical results the structure of the front is analysed using the jump size distribution introduced by hansen et al 1990

on a non equilibrium partition function for heat conduction the authors provide a physical interpretation for the lagrange multipliers associated with heat conduction in an information theoretical description of non equilibrium systems such an identification is based on the generalised gibbs equation of extended irreversible thermodynamics

scaling of overhangs in 1 1 dimensional directed processes in a gradient when a gradient is imposed in the control parameter of a directed process in a 1 1 dimensional directed process a front may occur the position of this front evolves in steps of size one in the direction opposite to the gradient and in steps of size equal to or larger than one in the other direction the larger steps are due to overhangs there is a critical value for the control parameter only if the first moment of the step size distribution in the direction of the gradient is not singular i e it takes on a finite value in the thermodynamic limit if there is a critical point this step size distribution is a power law at this point with an exponent larger than two and which the authors conjecture to be equal to 3 beta nu perpendicular to beta is the order parameter exponent and nu perpendicular to is the spatial correlation length exponent in a finite gradient there is a second exponent governing the upper cut off in the step size distribution this exponent is equal to nu perpendicular to 1 nu perpendicular to these two exponents govern the convergence of the position of the front towards the critical value of the control parameter the authors exemplify their discussion with directed site percolation and with two versions of a model originally designed to study catalytic reactions

the efficient determination of the percolation threshold by a frontier generating walk in a gradient the frontier in gradient percolation is generated directly by a type of self avoiding random walk the existence of the gradient permits one to generate an infinite walk on a computer of finite memory from this walk the percolation threshold pc for a two dimensional lattice can be determined with apparently maximum efficiency for a naive monte carlo calculation or n 12 for a square lattice the value pc 0 592745 or 0 000002 is found for a simulation of n 2 6 1011 total steps occupied and blocked perimeter sites the power of the method is verified on the kagome site percolation case

structure of backbone perimeters of percolation clusters the author studies perimeters of backbones of percolation clusters at the percolation threshold in two dimensions these perimeters are self avoiding for all lattices they simulate these perimeters by three independent processes measure the average size of them and study their variation with length they find the exponent characterising this variation to be 0 744 or 0 012 it seems likely that these perimeters are in the self avoiding walk universality class

p adic models of ultrametric diffusion constrained by hierarchical energy landscapes we demonstrate that the p adic analysis is a natural basis for the construction of a wide variety of models of ultrametric diffusion constrained by hierarchical energy landscapes a general analytical description in terms of the p adic analysis is given for a class of models two exactly solvable examples i e the ultrametric diffusion constrained by the linear energy landscape and the ultrametric diffusion with a reaction sink are considered we show that such models can be applied to both the relaxation in complex systems and the rate processes coupled to rearrangement of the complex surrounding

extrapolation of transfer matrix data for percolation and lattice animals by the romberg beleznay algorithm using beleznay s modified romberg algorithm the author re analyses the transfer matrix data for the two dimensional problems of derrida and stauffer 1985 he gets for percolation pc 0 59273 6 for the threshold of the square site problem nu 1 3330 7 for the correlation length exponent and omega 1 9 3 for the correction exponent in the case of lattice animals xc 0 246152 4 square lattice xc 0 19294 1 triangular lattice critical fugacity and nu 0 64062 10 radius exponent

non linear heat conduction in dense fluids in a dense fluid with very large heat flow j the thermal conductivity lambda lambda 0 lambda 2j2 an estimate of lambda 2 lambda 0 is made for hard spheres with mass and diameter appropriate to ar and density just below the gas solid transition this estimate indicates that fourier s law will hold in the form appropriate to low j if j or approximately 109 w m 2 and that the j dependence of pressure and internal energy is of the same order as that of lambda the calculation utilises an earlier derivation from the classical liouville equation which expresses lambda at high density in terms of the thermodynamic pressure the latter can be calculated with the aid of reciprocity and integrability conditions of extended non equilibrium thermodynamics which treats j as a state variable

p adic description of characteristic relaxation in complex systems this work is a further development of an approach to the description of relaxation processes in complex systems on the basis of the p adic analysis we show that three types of relaxation fitted into the kohlrausch williams watts law the power decay law and the logarithmic decay law are similar random processes inherently these processes are ultrametric and are described by the p adic master equation the physical meaning of this equation is explained in terms of a random walk constrained by a hierarchical energy landscape we also discuss relations between the relaxation kinetics and the energy landscapes

the bond bending model in three dimensions the author discusses the bond bending model in three dimensions he argues that for this model loops are very important in all length scales and the rigidity percolation is non local exhibiting similar behaviour to the central force model

multiparticle diffusion limited aggregation with strip geometry abstract two finite density versions of the multiparticle dla model have been investigated on strips of width l lattice units with periodic boundary conditions in model i single particles are added to the deposit at the position in which they first attempt to step onto the deposit and in model ii particles directly and indirectly connected to the deposit via nearest neighbor occupancy are added when they first contact the growing deposit for model i the growth of the deposit mass crosses over from t 1 2 dependence in the low density short time limit to m t in the high density long time limit for model ii a similar crossover is found but as the density approaches the percolation threshold the deposit mass is found to grow according to m t with having a value of about 4 3 these results help to resolve some apparent discrepancies between the results of earlier work on closely related models by voss and by meakin and deutch

elastic property of the self avoiding random walk calculates the end to end elastic susceptibility for the bond bending model on the self avoiding random walk the numerical result shows clearly that the central force term is irrelevant when the bond bending term is present the result also indicates that when the elastic constant of the bond bending term is set equal to infinity the elastic network is governed by the conductivity exponent of the random resistor network

specific heats of classical spin systems and inhomogeneous differential approximants derives accurate compact expressions for the high temperature specific heats of classical s infinity 3 vector spin systems on fcc bcc and sc lattices for pure heisenberg xy and ising like couplings respectively the analysis of the appropriate series expansions demonstrates the utility of inhomogeneous differential approximants and supports the estimate alpha h 0 21 or 0 04

information theoretic extended entropy for steady heat condition a phase space distribution for a far from equilibrium steady state in which thermodynamic temperature t and heat flux j are specified is calculated by maximising the information theoretic entropy when this distribution is used to calculate free energy f the latter is obtained as a function of the distribution moduli which in turn are calculated from a consistency condition as functions of j and t the coefficient of the o j2 term in f can be expressed in terms of the radial distribution function g2 and three particle distribution g3 a numerical estimate is made for a hard sphere model of liquid ar at 87 k and high density using the percus yevick solution for g2 and the kirkwood superposition for g3 the estimate confirms the conclusion of an earlier theory that non linear effects at liquid density are negligible except possibly for unrealistically large j fluctuation theory is invoked to make an estimate of the negligibly small o j4 term in f

some general relations between dressed self avoiding walks and percolation perimeters on lattices the author shows for some simple examples the relations between some recently proposed dressed self avoiding walks and site percolation connection between matching pairs and dressed walk generators is specified and some features of the corresponding bond percolation problem are given the case of surface generators for dressed self avoiding surfaces is briefly examined

surface growth on diluted lattices by a restricted solid on solid model an influence of diluted sites on surface growth has been investigated using the restricted solid on solid model it was found that with respect to equilibrium growth the surface width and the saturated width exhibited universal power law behaviors i e w ensuremath sim t ensuremath beta and w text sat ensuremath sim l ensuremath zeta regarding all cases with respect to the concentration of diluted sites x 1 ensuremath p with p being the occupation probability on each lattice site for xl x c 1 ensuremath p c p c being the percolation threshold the growth appeared to be similar to that of a regular lattice both in two and three dimensions for x x c the growth yielded nontrivial exponents which were different from those on a regular lattice in nonequilibrium growth a considerable amount of diluted sites x ensuremath le x c appeared to yield nonuniversal growth unlike the case of a regular lattice the cause of nonuniversal growth dynamics has been investigated considering the growth on a backbone cluster and on lattices constructed with periodically and randomly diluted subcells

ising n fold integrals as diagonals of rational functions and integrality of series expansions we show that the n fold integrals n of the magnetic susceptibility of the ising model as well as various other n fold integrals of the ising class or n fold integrals from enumerative combinatorics like lattice green functions correspond to a distinguished class of functions generalizing algebraic functions they are actually diagonals of rational functions as a consequence the power series expansions of the analytic at x 0 solutions of these linear differential equations derived from geometry are globally bounded which means that after just one rescaling of the expansion variable they can be cast into series expansions with integer coefficients we also give several results showing that the unique analytical solution of calabi yau odes and more generally picard fuchs linear odes with solutions of maximal weights are always diagonals of rational functions besides in a more enumerative combinatorics context generating functions whose coefficients are expressed in terms of nested sums of products of binomial terms can also be shown to be diagonals of rational functions we finally address the question of the relations between the notion of integrality series with integer coefficients or more generally globally bounded series and the modularity of odes

monte carlo experiments on percolation the influence of boundary conditions the slow convergence of the cluster size distribution observed in recent monte carlo simulations of two dimensional site percolation is shown to be due to the use of free boundary conditions when periodic boundaries are employed the convergence is improved considerably to the extent that the integrated size distribution for large typically 1010 site lattices is essentially flat over a range of cluster sizes spanning three orders of magnitude the existence of this plateau originally predicted by scaling theory leads to an improved estimate of the critical probability for the square lattice the observed correction to scaling is not readily characterised by a single additive term a fact which helps one to understand the wide range of published exponent estimates

self avoiding walk enumeration via the lace expansion we introduce a new method for the enumeration of self avoiding walks based on the lace expansion we also introduce an algorithmic improvement called the two step method for self avoiding walk enumeration problems we obtain significant extensions of existing series on the cubic and hypercubic lattices in all dimensions d 3 we enumerate 32 step self avoiding polygons in d 3 26 step self avoiding polygons in d 4 30 step self avoiding walks in d 3 and 24 step self avoiding walks and polygons in all dimensions d 4 we analyze these series to obtain estimates for the connective constant and various critical exponents and amplitudes in dimensions 3 d 8 we also provide major extensions of 1 d expansions for the connective constant and for two critical amplitudes

accessible external perimeters of percolation clusters the authors introduce a new definition of a cluster s accessible external perimeter which is different from the usual definition of the hull this external perimeter can be measured by probing adsorbent particles coming from the cluster s exterior the size of these particles determines which perimeter sites are accessible from the outside and hence one can define different external or accessible perimeters according to this size using a monte carlo simulation they measured the fractal dimensions of these accessible perimeters de for percolation clusters at the threshold on the square and the triangular lattices the fractal dimension of the accessible perimeter is found to depend on the size of the probing particles for several sizes which are larger than some lattice dependent threshold they find de approximately 4 3 which is significantly smaller than the hull dimension dh 7 4 found with smaller probing particles

relation between elastic and scalar transport exponent in percolation using simple variational considerations the author shows that the elastic critical exponent tau is related to the conductivity index t and to the correlation length exponent nu by tau or t 2 nu it is conjectured that this relation might be an equality

exact product form for the anisotropic simple cubic lattice green function the analytical properties of the lattice green function are investigated where n is an integer w is a complex variable and is a real parameter in the interval 0 in particular it is shown that g 2n n n w is a solution of a fourth order linear differential equation of the fuchsian type from this differential equation it is proved that g 2n n n w can be expressed in the hypergeometric form where and n denotes the pochhammer symbol this formula is valid for varying values of w in the neighbourhood of w provided that the argument function w does not take real values in the interval 1 the 2f1 product form is used to determine the asymptotic behaviour of g 2n n n w as n finally a five term linear recurrence relation is given for g 2n n n w

bond diluted rigid animals for central force model an exact enumeration of rigid clusters is analysed using pade approximants the number of bond diluted rigid clusters with n b bonds is found to grow like n theta b b 2 165 or 0 005 n b with theta b 0 988 a value different from that of site diluted rigid animals

scaling of overhangs appearing in fronts in higher dimensions directed percolation in higher dimensions serves as an example of how the distribution of overhangs gives information on the critical exponents of the system overhangs appear as jumps in the position of the front formed when a gradient in the control parameter is imposed along one of the spatial directions by analyzing the overhang distribution in d 1 dimensional directed percolation the authors determine the critical exponents beta and vperpendicular to for d2 in higher dimensions the overhang distribution is insensitive to the critical region

illumination a new method for studying 3d percolation fronts in a concentration gradient a new method of studying percolation fronts is introduced in situations where the concentration is not constant gradient percolation approach it consists in an efficient way of determining subsets of the percolation cluster which are exactly at the percolation threshold the method is applied to both lattice and continuum percolation in order to determine the percolation threshold in 3d more accurately two variations of this method illumination and projection seem to give a similar scaling behaviour at low gradients for the cubic lattice case leading to a new estimate for the percolation threshold pc 0 31173 or 0 00007 for continuum percolation of overlapping spheres the new estimate for the critical volume fraction is 0 291 or 0 002 and the value of the critical exponent alpha sigma for the width of the front scaled with the concentration gradient is very close to the one for the lattice case

attractors in fully asymmetric neural networks the statistical properties of the length of the cycles and of the weights of the attraction basins in fully asymmetric neural networks i e with completely uncorrelated synapses are computed in the framework of the annealed approximation which we previously introduced for the study of kauffman networks our results show that this model behaves essentially as a random map possessing a reversal symmetry comparison with numerical results suggests that the approximation could become exact in the large size limit

generation of percolation cluster perimeters by a random walk a type of self avoiding random walk whish generates the perimeter of two dimensional lattice percolation clusters is given the algorithm has been simulated on a computer yielding the mean perimeter length as a function of occupation probability

true self avoiding walks on a percolation cluster the authors investigate by monte carlo simulations the critical behaviour of true self avoiding walks tsaws on a percolation cluster performed very close to the percolation threshold specifically they generate tsaws on a site percolated incipient infinite cluster for various values of the self avoidance parameter g 0 they found that such walks exhibit critical behaviour different from that of ordinary self avoiding walks and also from that of random walks of no constraint the flory exponent obtained was v 0432 or 0 005 for all g 0 which agrees well with the flory type formula suggested by rammal 1984

concentration gradient approach to continuum percolation in two dimensions the author has investigated percolation properties of a system of overlapping discs randomly distributed with a gradient of concentration fractal properties and critical exponents of this system appear to be identical to their counterparts on a lattice this is in agreement with the universality of critical exponents of percolation the concentration gradient approach permits a precise calculation of the percolation threshold corresponding to a critical area fraction of 0 6766 or 0 0005 he has investigated the part of the external perimeter of the percolation cluster which is accessible to discs of various sizes the fractal dimension of this accessible perimeter is found to decrease abruptly as a function of the radius of invading discs from 1 75 or 0 02 to 1 35 or 0 02

invasion percolation between two sites in two three and four dimensions abstract the mass distribution of invaded clusters in non trapping invasion percolation between an injection site and an extraction site has been studied in two three and four dimensions this study is an extension of the recent study focused on two dimensions by araujo et al a d araujo t f vasconcelos a a moreira l s lucena j s andrade jr phys rev e 72 2005 041404 with respect to higher dimensions the mass distribution exhibits a power law behavior p m m it has been found that the index for p e p c p c being the percolation threshold of a regular percolation appears to be independent of the value of p e and is also independent of the lattice dimensionality when p e p c appears to depend marginally on the lattice dimensionality and the relation 1 being the exponent associated with cluster size distribution of a regular percolation via n s s appears to be valid

asymmetry in the percolation thresholds of fully penetrable disks with two different radii we perform simulations of continuum gradient percolation to measure the percolation threshold of systems of fully penetrable disks of two different radii as a function of volumetric proportion v the percolation threshold phic v lambda is approximately symmetric about v 0 5 for a fixed ratio lambda of the radii however the difference from symmetry is strongly statistically significant we also improve by an order of magnitude the measurement of the percolation threshold for disks of equal radius phic 0 676 347 5 6

correlated percolation and the correlated resistor network we present some exact results on percolation properties of the ising model when the range of the percolating bonds is larger than the nearest neighbours we show that the phase diagram for next nearest neighbour percolation can be exactly obtained from the nearest neighbour case which implies that the percolation threshold is still equal to the ising critical temperature in addition we present monte carlo calculations of the finite size behaviour of the correlated resistor network defined on the ising model the thermal exponent t of the conductivity that follows from it is found to be we observe no corrections to scaling in its finite size behaviour

classification of networks of automata by dynamical mean field theory dynamical mean field theory is used to classify the 65536 different networks of binary automata on a square lattice with nearest neighbour interactions application of mean field theory gives 700 different mean field classes which fall in seven classes of different asymptotic dynamics characterised by fixed points and 2 cycles

70 years of the watson integrals watson q j math oxford 10 266 276 1939 published in 1939 the evaluation of three integrals submitted to him which had arisen from a problem in physics van pepye in physica 5 465 482 1938 over the years these integrals have continued to occur in other aspects of physics such as random walk problems this article reviews these integrals and generalisations over the past 70 years

spreading and backbone dimensions of 2d percolation the author presents results of high statistics simulations of the spreading of 2d percolation and of backbones of 2d percolation clusters while the algorithm employed for the spreading is more or less standard for identifying backbones the author uses a new and very efficient recursive algorithm the author uses new critical exponents to test previous conjectures

applications of reciprocity to non linear extended thermodynamics kinetic equations abstract reciprocity relations and integrability conditions can be used in non linear extended irreversible thermodynamics to calculate forces and phenomenological coefficients uniquely in terms of parameters in kinetic equations derived from a model one assumes a set of n even variables i and corresponding i with an equation from the model for i independent thermodynamic forces terms are assumed known comparison of powers of in the model equation with corresponding powers in the canonical onsager casimir formalism yields equations to determine all quantities in the latter save the higher dependence of entropy which is affected by uncertainties in measurement by introducing a phenomenological ansatz one can relate these higher entropy terms to corresponding terms in the model equation a numerical illustration is made for a relaxing scalar structural parameter appropriate to a simple liquid

resistance of random walks unrestricted lattice random walks in which a unit conductor is placed along each bond traversed are considered the mean end to end resistance is studied as a function of the number of steps in the walk and the spatial dimension a critical scaling law is found whose exponent is consistently given by four different calculational schemes

nonlinear reciprocity and the maximum entropy formalism abstract the maximum entropy formalism calculates a phase space distribution p x which maximizes the information theoretic entropy subject to specified values of internal energy and state variables i and i i 1 n the latter are averages calculated from p of dynamical functions ai and ai ilai respectively where l is the liouville operator from p one extracts a fluctuation distribution g a v for the values of the a and a a kinetic equation for the fluctuation distribution has been derived by grabert using projection operators appropriate to a system in thermal contact with its surroundings and g is proposed as an ansatz for its solution from the first moments of the kinetic equation after introduction therein of g one extracts phenomenological equations for ai and hi whose terms can be grouped to exhibit onsager symmetry even when nonlinear terms are included the groupings can be affected exactly but not uniquely

efficient measurement of the percolation threshold for fully penetrable discs we study the percolation threshold for fully penetrable discs by measuring the average location of the frontier for a statistically inhomogeneous distribution of fully penetrable discs we use two different algorithms to efficiently simulate the frontier including the continuum analogue of an algorithm previously used for gradient percolation on a square lattice we find that c 0 676 339 0 000 004 thus providing an extra significant digit of accuracy to this constant some figures in this article appear in colour in the electronic version see www iop org

determination of the bond percolation threshold for the kagome lattice the hull gradient method is used to determine the critical threshold for bond percolation on the two dimensional kagome lattice and its dual the dice lattice for this system the hull walk is represented as a self avoiding trail or mirror model trajectory on the 3 4 6 4 archimedean tiling lattice the result one standard deviation of error is not consistent with previously conjectured values

cellular automata approach to non equilibrium diffusion and gradient percolation the authors propose a deterministic approach to lattice diffusion in two dimensions this method is implemented on a cellular automaton special purpose computer in order to study the properties of the interface of particles diffusing from a source to a sink fractal properties of the diffusion front of this non equilibrium process are compared with results from percolation theory the observed agreement indicates that diffusion fronts and gradient percolation coincide asymptotically and that the cellular automata method is a viable alternative to standard simulations for this class of problems

rigid clusters enumeration animal counting techniques suggest that the fraction of rigid animals among all animals with s sites on a triangular lattice varied roughly as s1 2 0 46 or 0 01 s

exact evaluation of the green functions for the anisotropic face centred and simple cubic lattices the analytic properties of the lattice green functions and are investigated where w1 w2 are complex variables and 1 2 are real parameters in the interval 0 in particular simple and direct methods are developed which enable one to evaluate g1 1 w1 and g2 2 w2 in terms of products of two complete elliptic integrals of the first kind kampe de feriet series are also used to derive new transformation formulae which give connections between g1 1 w1 and g2 2 w2

surface growth on percolation networks by a conserved noise restricted solid on solid growth model surface growth by the conserved noise restricted solid on solid model is investigated on diluted lattices i e on percolation networks that are embedded in two spatial dimensions the growth exponent and the roughness exponent are defined respectively by the mean square surface width via w 2 t t 2 and the mean square saturated width via w sat 2 l l 2 where l is the system size these are measured on both an infinite network and a backbone network and the results are compared with power counting predictions obtained using the fractional langevin equation while the monte carlo results on deterministic fractal substrates show excellent agreement with the predictions d h kim and j m kim phys rev e 84 011105 2011 the results on critical percolation networks deviate by 8 12 from these predictions

an algorithm for enumerating rigid clusters a more efficient algorithm for enumerating rigid clusters is presented using this algorithm the author has extended the series of site rigid animals enumerated by prunet and blanc 1986 by five more terms and has also calculated for the first time the exponent governing the growth of the radius of gyration for rigid animals

site percolation threshold for honeycomb and square lattices a new method of estimating the critical percolation threshold is proposed based on stauffer s cluster number scaling hypothesis and the universality with respect to lattice structure of the corresponding stauffer scaling function this method is illustrated by obtaining estimates of the site percolation threshold for the honeycomb lattice pc 0 6962 or 0 0006 and for the square lattice pc 0 5923 or 0 0007 the error bars or confidence limits of our estimates are substantially smaller than previous series estimates and for the honeycomb lattice would have to be multiplied by the factor 18 to include the value pc 2 12 recently proposed by kondor 1980 to be exact an additional result is r 4 95 or 0 15 where r lims to infinity rs and rs is the ratio of the cluster number scaling function at its maximum to its value at pc for clusters of size s

on the exact evaluation of the face centred cubic lattice green function the mathematical properties of the lattice green function open image in new window are investigated where w w 1 iw 2 lies in a complex plane which is cut from w 1 to w 3 and l 1 l 2 l 3 is a set of integers with l 1 l 2 l 3 equal to an even integer in particular it is proved that g 2n 0 0 w where n 0 1 2 is a solution of a fourth order linear differential equation of the fuchsian type with four regular singular points at w 1 0 3 and it is also shown that g 2n 0 0 w satisfies a five term recurrence relation with respect to the integer variable n the limiting function g 2n 0 0 w 1 equiv lim epsilon rightarrow0 g 2n 0 0 w 1 mathrm i epsilon g mathrm r 2n 0 0 w 1 mathrm i g mathrm i 2n 0 0 w 1 nonumber where w 1 1 3 is evaluated exactly in terms of 2 f 1 hypergeometric functions and the special cases g 2n 0 0 0 g 2n 0 0 1 and g 2n 0 0 3 are analysed using singular value theory more generally it is demonstrated that g l 1 l 2 l 3 w can be written in the form open image in new window where open image in new window are rational functions of the variable k k and e k are complete elliptic integrals of the first and second kind respectively with k 2 equiv k 2 w 1 over2 2 over w biggl 1 1 over w biggr 3 over2 1 over2 biggl 1 1 over w biggr biggl 1 1 over w biggr 3 over2 biggl 1 3 over w biggr 1 over2 nonumber and the parameter is defined as xi equiv xi w biggl 1 sqrt 1 3 over w biggr 1 biggl 1 sqrt 1 1 over w biggr nonumber this result is valid for all values of w which lie in the cut plane the asymptotic behaviour of g 2n 0 0 w 1 and g 2n 0 0 w 1 as n is also determined in the final section of the paper a new 2 f 1 product form for the anisotropic face centred cubic lattice green function is given