critical slowing down in a randomly biased percolating system at percolation threshold using computer simulation a random walk motion is studied in a random percolating system at percolation threshold in the presence of a random field the rms displacement is found to decrease systematically as a function of field strength and shows a non universal power law dependence on it the long time behaviour seems similar to the critical slowing down recently predicted in an analogous magnetic system the spectral dimensionality varies with the field strength and is found to be lees than

new series expansion data for surface and bulk resistivity and conductivity in two dimensional percolation low density series expansions are obtained for random resistor networks the expansions are obtained on the square lattice to order 18 and on the triangular lattice to order 14 bulk and surface expansions are given for both resistive and conductive susceptibilities the balance of the evidence obtained from analysing these series is in favour of the value for the critical exponent of the resistance scale and supports the existence of only a single such scale for bulk and surface susceptibilities this value is in agreement with earlier monte carlo work

disorder induced transport i simple cubic lattice using a vax 11 780 a random walk motion is studied in a random field disordered system a non monotonic transport behaviour is observed as a function of local field intensity in contrast to the random walk motion in random percolating fractals the root mean square displacement here increases as a function of time faster than that of diffusion a crossover from diffusion to disorder induced transport is discussed and its spectral dimensionality is estimated

random walk on fractals numerical studies in two dimensions monte carlo calculations are used to investigate some statistical properties of random walks on fractal structures two kinds of lattices are used the sierpinski gasket and the infinite percolation cluster in two dimensions among other problems the authors study i the range rn of the walker number of distinct visited sites during n steps average value sn variance sigma n and asymptotic distribution ii renewal theory return to the original site probability of return p0 n mean number of returns nu n the probability distribution of the walker position p n r after n steps is discussed the asymptotic behaviour n 1 of these quantities exhibits power laws with associated exponents the numerical values of these exponents are in good agreement with recent theoretical predictions alexander and orbach 1982 rammal and toulouse 1982

branched polymer approach to the structure of lattice animals and percolation clusters the authors treat the percolation problem as a solution of randomly branched chains and argue that such chains are swollen by a three body repulsion acting only within a connected cluster this type of repulsion should not be subject to the screening effects which inhibit ordinary repulsions the repulsion becomes relevant below six dimensions they estimate its effect on the size of a cluster in analogy with flory s argument for the swelling of linear chains above six dimensions the cluster size scales as mass to the 1 4 power as with isolated branched chains in high dimensions

fractal dimensionality and the number of visited sites of the ant in the labyrinth monte carlo studies in three dimensions confirm the hypothesis of toulouse and rammal that the number of distinct sites visited by a random walk on a random network at its percolation threshold varies as time 23

speculations on self avoiding surfaces in fractals a mean field treatment the authors estimate the exponents characterising the self avoiding surfaces using an approximation in the framework of a flory type theory they find for planar self avoiding surfaces embedded randomly in a fractal of dimensionality d nu 3 4 d for random surfaces of fractal dimension d embedded in a euclidean space of dimensionality d nu 3 2d d 2 and for fractal surfaces embedded in a structure of fractal dimensionality d nu 3 2d d 2

geometrical arguments against the alexander orbach conjecture for lattice animals and diffusion limited aggregates geometrical arguments are given which suggest that the alexander orbach conjecture 1982 does not hold for lattice animals in d 2 and for the diffusion limited aggregates for large dimensions

size dependent diffusion coefficient in a myopic random walk on a strip abstract we consider a random walk in discrete time n 0 1 2 on a square lattice of finite width in the y direction i e j m j z m 1 2 3 n a myopic walker at j 1 or j n jumps with probability 1 3 to any of the available nearest neighbor sites at the end of a time step this couples the motions in the x and y directions and leads to several interesting features including a coefficient of diffusion in the x direction that depends on the transverse size n of the strip explicit solutions for x2n and the lateral variance y2n are given for small values of n a closed form expression is obtained for the discrete laplace transform of x2n for general n the asymptotic behaviors of x2n and y2n are found the corrections falling off exponentially with increasing n the results obtained are generalized to a myopic random walk in d dimensions and it is shown that the diffusion coefficient has an explicit geometry dependence involving the surface to volume ratio this coefficient can therefore serve as a probe of the geometry of the structure on which diffusion takes place

diffusion in three dimensional random systems at their percolation thresholds extensive monte carlo simulations of theant in the labyrinth problem on randoml l l simple cubic lattices are performed forl up to 960 on a cray ymp supercomputer the exponentk for the rms displacementr witht inr t k is found to bek 0 190 0 003 as a second approach large percolation clusters with chemical shells up to 300 are generated on a simple cubic lattice at criticality the diffusion equation is then solved by using the exact enumeration technique the corresponding critical exponentd w is found to be 1 d w 0 250 0 003

effective renormalization group algorithm for transport in oil reservoirs abstract a real space renormalization group algorithm is used to evaluate macroscopic permeabilities the algorithm was tested on many different distributions with large spatial correlations and anisotropies and the results are very close to the expected exact values

a new method for the calculation of the conductivity of inhomogeneous systems a new method for computing the conductivity of random irregular resistor networks is developed this method is a generalization of the transfer matrix technique proposed by derrida and vannimenus for regular 2d and 3d lattices at the same time for large systems the method presented in this paper is more efficient than the transfer matrix technique to demonstrate the method it is applied to a cubic lattice at the percolation threshold and away from it the conductivity has been found for lattices with size up to 324 3 the ratio between the conductivity exponent t and the correlation length exponent was estimated to be t 2 315 in good agreement with the literature data

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

perturbative treatments and learning techniques the replica trick rt of statistical mechanics has become a very useful tool for the investigation of complex systems in general and in particular for studying learning and generalization processes in neural networks nn 1 2 the trick overcomes the difficulty of performing an ensemble average of the logarithm of the partition function z by that of averaging z over n replicas of the original network with n a very large integer for feedforward single layered nn 3 4 the rt ansatz has been proved to be a valuable tool for the learning of a rule on the basis of a suitable set of examples in particular the full quenched theory has been carefully studied within the framework of a replica symmetry approximation 5 however as the temperature drops symmetry breakings may invalidate this approximation so that an approach to the full quenched theory that incorporates disorder effects due to improper examples may be of some interest in the present effort we wish to study in the spirit of a second order approximation noise effects in the training set unavoidable in any realistic setting the noise will here be the result of letting just part of the examples to be produced by the perceptron teacher pt the rest are to be randomly selected bad examples two types of situations are to be confronted learnable and unlearnable rules 6 for the former there is at least a vector in the concomitant weight space that can learn the rule in an exact fashion the latter arises mostly in cases of architectural mismatch in such a situation the training error can never vanish the question to be answered is can a perceptron trained under these circumstances correctly respond to queries posed by a pt in other words is the rule underlying the good examples a learnable one we will show here that these questions can be adequately dealt with the paper is organized as follows section ii is devoted to a brief recapitulation of basic concepts concerning the rt while sec iii deals with the thermodynamics of the situation that interests us here a second order high temperature approximation is derived boolean perceptrons with ising weights on the learning curves are the subject of sec iv finally some conclusions are drawn in sec v ii the replica method

monte carlo simulation of long time percolation diffusion on lattices above the threshold monte carlo simulations of percolation diffusion at and below the percolation threshold give results in accordance with theory above the percolation threshold however this is not the case it was thought that above the threshold a well defined crossover point separated the anomalous and classical diffusion regimes with the classical diffusion coefficient having the same critical exponent as the lattice conductivity but monte carlo simulations failed to confirm this expected behaviour analysis of new monte carlo results presented here for the square lattice shows that percolation diffusion is classical only strictly asymptotically instead of a crossover the approach to classical behaviour is better described as a very slow relaxation accounting for this relaxation enables the critical behaviour of percolation diffusion to be confirmed with a corresponding critical exponent

markov matrix analysis of random walks in disordered continuous media abstract we study by a markov matrix analysis of the equivalent random walks the dynamic properties of continuous media consisting of both correlated and uncorrelated equal size spheres we employ a blind ant random walk model using the rule that a walker jumps among centers of the directly connected spherical particles on an infinite network the dominant eigenvalues and eigenvectors of the transition probability matrix of the random walks are calculated yielding estimates of the spectral dimension d s and the fractal dimension d w of random walks on the continuous network we find that for the present model the estimates are very close to the corresponding lattice percolation values though only after the finite size effects have been carefully taken into account we also show that the finite size scaling of the largest nontrivial eigenvalues holds for our model with the same exponents as for the lattice percolation

robust algorithm for random resistor networks using hierarchical domain structure in this study we discuss methods for solving random resistor networks and similar problems we discuss the node elimination method and we demonstrate its equivalence to the gaussian elimination scheme finding a good elimination order which makes the method highly efficient the transfer matrix method is shown to be a special case of the node elimination method with an ordering that is far from optimal we compare the performance of these exact methods with a state of the art conjugate gradient solver in general the node elimination method is the faster method

characterization and quantification of the cluster hopping mechanism responsible for transport in two component random networks a cluster hopping mechanism for transport in two component random networks below pc is proposed and shown to account for observed conductivity exponents key structures forming paths of least resistance are identified a novel computationally efficient method for determining exponents results

spiral growing self avoiding walk a new universality class the author introduces the concept of a spiral growing self avoiding walk sgsaw by imposing privman s spiralling constraint ibid vol 16 p l571 1983 on the growing self avoiding walk introduced by majid et al phys rev lett vol 52 p 1257 1984 and lyklema and kremer ibid vol 17 p l691 1984 he studies the critical properties of the sgsaw using monte carlo simulation the exponent nu associated with the average radius of gyration sn is 0 8 on the other hand the critical exponent nu r associated with the end to end distance rn is much less than nu the latter is in sharp contrast to the result nu nu r for the usual saw directed gsaws are shown to behave like directed usual saws

current distribution in the three dimensional random resistor network at the percolation threshold we study the multifractal properties of the current distribution of the three dimensional random resistor network at the percolation threshold for lattices ranging in size from 8 3 to 80 3 we measure the second fourth and sixth moments of the current distribution finding e g that t ensuremath nu 2 282 5 where t is the conductivity exponent and ensuremath nu is the correlation length exponent textcopyright 1996 the american physical society

conductivity exponent and backbone dimension in 2 d percolation abstract we present high statistics simulations for 2 d percolation clusters in the bus bar geometry at the critical point for site and for bond percolation we measured their backbone sizes and electrical conductivities for all sets of measurements we find large corrections to scaling most of which do not seem to be described by single powers using single power terms for the corrections to scaling of the backbone masses we would obtain fractal dimensions which are different for site and bond percolation while the correct result is d b 1 6432 0 0008 for both for the conductivity the corrections to scaling are strongly non monotonic for bond percolation the exponent t t is measured as 0 9826 0 0008 in disagreement with the alexander orbach and other conjectures

topological proof for the alexander orbach conjecture this paper advocates an unconventional analytical approach to studying the fractal geometry of percolation at the threshold which is based on the most general methods of the differential topology our particular interest concentrates on the alexander orbach ao conjecture j phys france lett 43 l625 1982 which assigns the universal mean field value 4 3 to the spectral fractal dimension stackrel ifmmode tilde else fi d at the percolation threshold for all embedding euclidean dimensions n greater than one i e ng 2 using the topological arguments we show that the ao conjecture might be improved for the relatively low embedding dimensions 2l nl 5 for which the analytical result stackrel ifmmode tilde else fi d 1 327 ifmmode pm else textpm fi 0 001 is proposed instead of the original ao estimate 4 3 meanwhile we assume that the exact value stackrel ifmmode tilde else fi d 4 3 holds for all ng 6 as it follows directly from the well known mean field theory the improved value of stackrel ifmmode tilde else fi d ensuremath approx 1 327 for 2l nl 5 is obtained from an analysis of the basic topological properties of the percolating fractal sets at the threshold of percolation we show that these properties could be investigated fruitfully with the introduction of the concept of the fractal manifold which might serve as an effective instrument when considering the topology of the fractal objects our results indicate that the proposed value of stackrel ifmmode tilde else fi d ensuremath approx 1 327 for the spectral dimension at the percolation threshold has the fundamental topological origin related to the most general features of the fractal geometry of percolation at criticality we argue that the constraint 2l nl 5 on the topological dimension n of the embedding euclidean space is the direct consequence of the famous whitney theorem which establishes the embedding properties of manifolds from the viewpoint of their dimensionality a simple topological condition that identifies the threshold of percolation is obtained for 2l nl 5 the particular topological restrictions implied throughout the present study are discussed and the important issue of contractibility of the fractal manifolds is pointed out

biased random walks and propagation failure the critical value of the reaction rate able to sustain the propagation of an invasive front is obtained for general non markovian biased random walks with reactions from the hamilton jacobi equation corresponding to the mean field equation we find that the critical reaction rate depends only on the mean waiting time and on the statistical properties of the jump length probability distribution function and is always underestimated by the diffusion approximation if the reaction rate is larger than the jump frequency invasion always succeeds even in the case of maximal bias numerical simulations support our analytical predictions

scaling theory for the dynamic density profile at criticality a scaling description for the dynamic density profile of kinetic clusters formed by the sites visited during random walks executed on critical percolation clusters is proposed the analysis is done for both euclidean and fractal spaces various percolative diffusion laws are correctly recovered from this dynamic density profile

use of power averaging for quantifying the influence of structure organization on permeability upscaling in on lattice networks under mean parallel flow with the local permeability distribution variance s 2 is nonnegligible but remains small it is equal to 0 09 for sparse networks and 0 14 for dense networks representing 4 5 and 7 respectively of the full possible range of w values power averaging is not strictly valid but gives an estimate of upscaling at a few percent here w depends slightly on the local permeability distribution shape beyond its variance but mostly on the morphological network structures most of the morphological control on upscaling for on lattice structures is local and topological and can be explained by the dependence on the average number of neighbor by points effective coordination number within the flowing structure backbone

there is more to be learned from the lorentz model the classical lorentz model for charged noninteracting point particles in a perpendicular magnetic field is reconsidered in 2d we show that the standard boltzmann equation is not valid for this model even in the grad limit we construct a generalized boltzmann equation which is and solve the corresponding initial value problem exactly by an independent calculation we find the same solution by directly constructing the green function from the dynamics of the model in the grad limit from this solution an expression for the diffusion tensor valid for arbitrary short range forces is derived for hard disks we calculate the diffusion tensor explicitly away from the grad limit a percolation problem arises we determine numerically the percolation threshold and the corresponding geometric critical exponents the numerical evidence strongly suggests that this continuum percolation model is in the universality class of 2d lattice percolation although we have explicitly determined a number of limiting properties of the model several intriguing open problems remain

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

critical dynamics at the percolation threshold dilution induced criticality is discussed using exact and scaling descriptions a simple exact argument is used to obtain the scaling form for the density of states in dilute chains for higher dimensional systems it is shown that the condition for the existence of a fraction edge is equivalent to the violation of a scaling relation between the fracton dimension of the infinite cluster d and various static exponents in addition the result dd df is obtained for the fracton dimensionality of the finite clusters near the percolation threshold df is the fractal dimension of the infinite cluster

spreading of percolation in three and four dimensions the authors present monte carlo simulations of the spreading of bond percolation on three and four dimensional simple hyper cubic lattices the algorithm is the same as that applied previously in two dimensions with the spreading proceeding from a hyperplane they find the spreading dimension to be d 1 82 or 0 02 three dimensional and d 1 88 or 0 03 four dimensional also they obtain values of the percolation probability and of the static exponents with errors at least comparable to the best values from the literature

universality of 3d percolation exponents and first order corrections to scaling for conductivity exponents abstract by using a fast nested dissection algorithm we compare the results of finite size scaling at p c and of p scaling l const on large cubic random resistor networks up to 500 500 500 the p scaling for conductivity of both site and bond networks leads to an exponent t 2 00 1 the finite size scaling leads to the ratio of this conductivity exponent to the coherence length exponent t 2 283 3 combining these results we estimate 0 876 6 in excellent agreement with a value proposed by ballesteros et al the first order correctional exponent is found to be 1 0 2

low concentration series in general dimension we discuss recent work on the development and analysis of low concentration series for many models the recent breakthrough in the extremely efficient no free end method of series generation facilitates the derivation of 15th order series for multiple moments in general dimension the 15th order series have been obtained for lattice animals percolation and the edwards anderson ising spin glass in the latter cases multiple moments have been found from complete graph tables through to 13th order general dimension 13th order series have been derived for the resistive susceptibility the moments of the logarithms of the distribution of currents in resistor networks and the average transmission coefficient in the quantum percolation problem 11th order series have been found for several other systems including the crossover from animals to percolation the full resistance distribution nonlinear resistive susceptibility and current distribution in dilute resistor networks diffusion on percolation clusters the dilute ising model dilute antiferromagnet in a field and random field ising model and self avoiding walks on percolation clusters series for the dilute spin 1 2 quantum heisenberg ferromagnet are in the process of development analysis of these series gives estimates for critical thresholds amplitude ratios and critical exponents for all dimensions where comparisons are possible our series results are in good agreement with bothe expansion results near the upper critical dimension and with exact results when available in low dimensions and are competitive with other numerical approaches in intermediate realistic dimensions

precise calculation of the dynamical exponent of two dimensional percolation we report numerical data obtained on the special purpose computer percola for the exponent tilde t of the electrical conductivity of 2d percolation the extrapolation yields tilde t 0 9745 pm 0 0015 and a correction to the scaling exponent 1 2 0 2

series expansion evidence supporting the alexander orbach conjecture in two dimensions the authors extend the series expansion of fisch and harris 1978 for the resistive susceptibility chi r p by a further six terms on the square lattice this leads to a more precise estimate of the corresponding exponent gamma r 3 65 or 0 02 they also obtain the exact relation chi r p 2e tau mod f where tau is the sum of the relaxation times for charge diffusion on the cluster containing the origin and the expectation value is subject to the condition that the latter is finite a known scaling relation for the fracton dimension df in terms of gamma r and the static exponents is derived without the usual reference to the infinite cluster using the estimate of gamma r they find df 1 334 or 0 007 which is consistent with the ao conjecture df 4 3 they also note that gamma r is the same for directed and undirected percolation to within the accuracy of the calculations

a direct electrifying algorithm for backbone identification this paper proposes a new algorithm for identifying backbones in the application of percolation theory this algorithm is based on the current carrying definition of backbone and is carried out on the predetermined spanning cluster it is fairly easy to implement and further parallelize the efficiency is enhanced by the fact that the conductivity of a percolating system can be obtained in the same processing of backbone identification the critical exponents of backbone mass red bonds sites and conductivity obtained by this algorithm are in very good agreement with the existing results

renormalization group methods for flow in oil reservoirs abstract oil reservoirs are described as networks of bonds with random permeabilities which exhibit fractal features real space renormalization group methods give very accurate estimates of the reservoir s global effective permeability

a precise characterization of three dimensional percolating backbones the backbones of three dimensional critical percolation clusters are extracted and the fractal dimension is accurately calculated using various mass scaling and box counting techniques to be dfb 1 855 or 0 015 the eigenvalue spectrum of the brownian transition probability matrix is studied numerically which results in the estimates dwb 3 13 or 0 03 for the walk dimension and dsb 1 18 or 0 01 for the spectral dimension of the backbone independent calculation of all three quantities allows the conductivity exponent mu to be determined and also allows the validity of the scaling relation ds 2df dw proposed by alexander and orbach 1982 to be tested on the backbone

numerical studies of critical percolation in three dimensions presents results of high statistics simulations of the spreading of 3d percolation close to the critical point the main results are i a more precise estimate of the spreading resp chemical distance dimension ii a novel heuristic scaling theory for percolation near surfaces and iii more precise values for critical surface and edge exponents in addition the author verifies the very precise results of ziff and stell 1988 for the percolation thresholds and studies in detail corrections to scaling the latter is made possible by simulating bond and site percolation in a number of different geometries in addition in most cases one could measure more than one observable during each simulation which provided quite severe internal consistency constraints in some cases the author found quite important deviations from scaling which are not easily described by a single power or logarithmic correction term

on random walks in percolation models random walks on percolation clusters of any dimension consisting of both directed and undirected bonds are studied results conjectured to be exact and supported by monte carlo simulation are deduced for the critical parameters of the problem it is shown that in the undirected case of the model the probabilities of finding the particle in the sites of a finite cluster have a uniform limit distribution

diffusion on lattice animals and percolation clusters a renormalisation group approach a position space renormalisation group method is developed to study the scaling properties of random walks on fractals calculations in two dimensions for site lattice animals and site percolation clusters yield numerical results for the critical exponent defined by the asymptotic behaviour of the root mean square displacement of the walker the renormalisation group results are consistent with recent monte carlo calculations

a transfer matrix approach to random resistor networks introduces a transfer matrix formulation to compute the conductance of random resistor networks the authors apply this method to strips of width up to 40 and use finite size scaling arguments to obtain an estimate for the conductivity critical exponent in two dimensions t 1 28 or 0 03

field induced drift and trapping in percolation networks the authors study the effects of a bias producing external field on a random walk on the infinite cluster in the percolation problem there are two competing physical effects drift and trapping which result in a drift velocity nu which rises and then falls as the field increases they study these effects on a one dimensional lattice with random length branches and on the diluted bethe lattice they calculate nu in the first model with a maximum allowed length for each branch in the limit that this cutoff length becomes infinite they find that the velocity vanishes identically above a finite critical value of the field for the bethe lattice they derive an upper bound on the critical field which varies as p pc 1 2 as the percolation concentration pc is approached in the one dimensional model they also investigate the anomalous regime in which the velocity vanishes they discuss the distribution of steady state times required to traverse n sites and find that it can be described in terms of a stable distribution of index x with superimposed oscillations the index x of the stable distribution is given by l xi where xi is a characteristic branch length and l is a bias induced length which describes the exponential buildup of the steady state density of particles towards the end of a branch

tree adjoining grammars for rna structure prediction in this paper we are concerned with identifying a subclass of tree adjoining grammars tags that is suitable for the application to modeling and predicting rna secondary structures the goal of this paper is twofold for the purpose of applying to the rna secondary structure prediction problem we first introduce a special subclass of tags and develop a fast parsing algorithm for the subclass together with some of its language theoretic characterizations then based on the algorithm we develop a prediction system and demonstrate the effectiveness of the system by presenting some experimental results obtained from biological data where free energy evaluation selection for parse trees is incorporated into the algorithm

corrections to scaling for diffusion exponents on three dimensional percolation systems at criticality recent results of monte carlo simulations of the ant in the labyrinth method in three dimensional percolation lattices are reanalyzed in the light of more accurate corrections to scaling ansatz motivated by inconsistent results that have appeared in the literature the results are observed to be sensitive to the form of the scaling correction terms using a single correction term we estimate the valuek 0 197 0 004 for the anomalous diffusion exponent at criticality when two correction terms are included k 0 200 0 002 is obtained these new estimates are consistent with known theoretical bounds with recent series expansion results and with numerical calculations of the conductance of random resistor networks above criticality

random walks on invasion percolation clusters results are presented of numerical simulations of random walks on invasion percolation clusters in two and three dimensions including simulations on continuum percolation clusters at threshold walks of up to 105 steps were taken the results indicate that the fracton dimension of invasion percolation clusters both on regular lattices and in the continuum is the same within numerical errors as the fracton dimension of ordinary percolation clusters at threshold

discrete network models for the low field hall effect near a percolation threshold theory and simulations the critical behavior of the weak field hall effect near a percolation threshold is studied with the help of two discrete random network models many finite realizations of such networks at the percolation threshold are produced and solved to yield the potentials at all sites a new algorithm for doing that was developed that is based on the transfer matrix method the site potentials are used to calculate the bulk effective hall conductivity and hall coefficient as well as some other properties such as the ohmic conductivity the size of the backbone and the number of binodes scaling behavior for these quantities as power laws of the network size is determined and values of the critical exponents are found

possibly exact fractal dimensions from conformal invariance based on an equivalence between the q state potts model and conformally invariant field theories the author speculates that all fractal dimensionalities of two dimensional percolation are given by the formula d 100 x2 48 where x is an integer the same formula should also apply to self avoiding walks and lattice animals this approach is extended to cluster weighted percolation his conjecture seems to agree with previous exact and numerical results except for the backbone exponent and possibly for lattice animals

exponent inequalities for the bulk conductivity of a hierarchical model the bulk conductivity p of the bond lattice in d is considered where the bonds have conductivity 1 with probabilityp or e 0 with probability 1 p various representations of the derivatives of p are developed these representations are used to analyze the behavior of p for e 0 near the percolation thresholdp c when the conducting backbone is assumed to have a hierarchical node link blob nlb structure this model has loops on arbitrarily many length scales and contains both singly and multiply connected bonds exact asymptotics of frac d 2 sigma dp 2 for the nlb model are proven under some technical assumptions the proof employs a novel technique whereby frac d 2 sigma dp 2 for the nlb model with e 0 andp nearp c is computed using perturbation theory for p for two and three component resistor lattices aroundp 1 with a sequence of e s converging to 1 as one goes deeper in the hierarchy these asymptotics establish convexity of p for the nlb model nearp c and that its critical exponentt obeys the inequalities 1 t 2 ford 2 3 while 2 t 3 ford 4 the upper boundt 2 ind 3 which is realizable in the nlb class virtually coincides with two very recent numerical estimates obtained from simulation and series expansion for the original model

diffusion on percolation clusters at criticality the concept of fractal dimensionality is used to study the problem of diffusion on percolation clusters the authors find from monte carlo simulations that the fractal dimensionality of a random walk on a critical percolation cluster in three dimensional space is d 3 3 or 0 1 where the size of the cluster is restricted to be larger than the span of the walk and is d 3 9 or 0 1 for a walk on clusters not subject to this restriction for two dimensional space they find d approximately d approximately 2 7 or p0 1 the exponent d and d is related to the scaling of the average length r of n steps via rd varies as n the fracton dimensionality which is related to the density of states was found to be 1 26 or 0 1 these results are in good agreement with the predictions of alexander and orbach 1982

optimal storage properties of neural network models the authors calculate the number p alpha n of random n bit patterns that an optimal neural network can store allowing a given fraction f of bit errors and with the condition that each right bit is stabilised by a local field at least equal to a parameter k for each value of alpha and k there is a minimum fraction fmin of wrong bits they find a critical line alpha c k with alpha c 0 2 the minimum fraction of wrong bits vanishes for alpha alpha c k the calculations are done using a saddle point method and the order parameters at the saddle point are assumed to be replica symmetric this solution is locally stable in a finite region of the k alpha plane including the line alpha c k but there is a line above which the solution becomes unstable and replica symmetry must be broken

the space of interactions in neural network models the typical fraction of the space of interactions between each pair of n ising spins which solve the problem of storing a given set of p random patterns as n bit spin configurations is considered the volume is calculated explicitly as a function of the storage ratio alpha p n of the value kappa 0 of the product of the spin and the magnetic field at each site and of the magnetisation m here m may vary between 0 no correlation and 1 completely correlated the capacity increases with the correlation between patterns from alpha 2 for correlated patterns with kappa 0 and tends to infinity as m tends to 1 the calculations use a saddle point method and the order parameters at the saddle point are assumed to be replica symmetric this solution is shown to be locally stable a local iterative learning algorithm for updating the interactions is given which will converge to a solution of given kappa provided such solutions exist

scaling behavior of permeability and conductivity anisotropy near the percolation threshold we use the finite size scaling method to estimate the critical exponent that characterizes the scaling behavior of conductivity and permeability anisotropy near the percolation thresholdpc here is defined by the scaling lawkl kt 1 p pc wherekt andkt are the conductivity or permeability of the system in the direction of the macroscopic potential gradient and perpendicular to this direction respectively the results are d 2 0 819 0 011 and d 3 0 518 0 001 we interpret these results in terms of the structure of percolation clusters and their chemical distance we also compare our results with the predictions of a scaling theory for due to straley and propose that d 2 t b wheret is the critical exponent of the conductivity or permeability of the system and b is the critical exponent of the backbone of percolation clusters

on correction to scaling for two and three dimensional scalar and vector percolation a reanalysis of the resistancer of two and three dimensional superconducting networks at the percolation thresholdp c together with the previous results for the elastic modulik of such networks shows that there is aunified description of finite size scaling for scalar and vector transport properties of percolation systems for a network of linear sizel atp c and forboth scalar and vector percolation inboth two and three dimensions k andr scale withl asl x a 1 a 2 lnl 1 a 3 l 1 wherex is the ratio of the associated critical exponent ofk orr and the correlation length exponentv of percolation although our estimates ofx for the resistance of percolation networks are consistent with the previous results they do indicate that inboth two and three dimensions and forboth scalar and vector percolation the leading nonanalytic correction to scaling exponent iszero from a reanalysis of data on diffusion on percolation clusters atp c we propose that such correction to scaling terms are a general property ofdynamics of percolation clusters we also suggest that for two dimensional percolation the conductivity exponentt and the superconductivity exponents are given by s t v 4 187 144 1 2986 and the elasticity exponentf is given byf t 2v 571 144 3 9652 where is the exponent of the strength of the infinite percolation cluster

single random walker on disordered lattices random walks on square lattice percolating clusters were followed for up to 2 10 steps the mean number of distinct sites visited sn gives a spectral dimension of d s 1 30 5 0 03 consistent with superuniversality d s 4 3 but closer to the alternative ds 182 139 based on the low dimensionality correction simulations are also given for walkers on an energetically disordered lattice with a jump probability that depends on the local energy mismatch and the temperature an apparent fractal behavior is observed for a low enough reduced temperature above this temperature the walker exhibits a crossover from fractal to euclidean behavior walks on two and three dimensional lattices are similar except that those in three dimensions are more efficient

real space renormalisation group method for the dilute gaussian model and its relation to the random walk problems near the percolation threshold an isomorphism between the gaussian model in the magnetic field and the time dependent conduction diffusion problem permits the author to develop a real space renormalisation group method for the diluted version of this model by which all single cluster results of the phenomenological scaling theory of gefen aharony and alexander 1983 are correctly reproduced an extension of this model to the case of strong bias then provides some restrictions on the validity of their results the result for the density of states which the author reproduces in the form predicted by alexander and orbach 1982 is subject to the same kind of restrictions

comment on transport processes on fractal structures the scaling exponents for the depleted sierpinski gasket derived by havlin and ben avraham ibid vol 16 p l483 7 1983 are shown to differ sharply from those derived by the method of given and mandelbrot ibid vol 16 p l565 9 1983 the discrepancy is analysed and it is shown that the argument used by havlin and ben avraham is incorrect

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

diffusion on random systems above below and at their percolation threshold in two and three dimensions a detailed monte carlo study is presented for classical diffusion random walks on random l l triangular and l l l simple cubic lattices with l up to 4096 and 256 respectively the speed of a cyber 205 vector computer is found to be about one order of magnitude larger than that of a usual cdc cyber 76 computer to reach the asymptotic scaling regime walks with up to 10 million steps were simulated with about 10 jl steps in total for l 256 at the percolation threshold we review and extend the dynamical scaling description for the distance traveled as function of time the diffusivity above the threshold and the cluster radius below earlier discrepancies between scaling theory and computer experiment are shown to be due to insufficient monte carlo data the conductivity exponent is found to be 2 0 0 2 in three and 1 28 0 02 in two dimensions our data in three dimensions follow well the finite size scaling theory below the threshold the approach of the distance traveled to its asymptotic value is consistent with theoretical speculations and an exponent 2 5 independent of dimensionality the correction to scaling exponent at pc seems to be larger in two than in three dimensions

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

classifying rna pseudoknotted structures computational prediction of the minimum free energy mfe secondary structure of an rna molecule from its base sequence is valuable in understanding the structure and function of the molecule since the general problem of predicting pseudoknotted secondary structures is np hard several algorithms have been proposed that find the mfe secondary structure from a restricted class of secondary structures in this work we order the algorithms by generality of the structure classes that they handle we provide simple characterizations of the classes of structures handled by four algorithms as well as linear time methods to test whether a given secondary structure is in three of these classes we report on the percentage of biological structures from the pseudobase and gutell databases that are handled by these three algorithms

on multiple context free grammars abstract multiple context free grammars mcfg s is a subclass of generalized context free grammars introduced by pollard 1984 in order to describe the syntax of natural languages the class of languages generated by mcfg s called multiple context free languages or shortly mcfl s properly includes the class of context free languages and is properly included in the class of context sensitive languages first the paper presents results on the generative capacity of mcfg s and also on the properties of mcfl s such as formal language theoretic closure properties next it is shown that the time complexity of the membership problem for multiple context free languages is o ne where n is the length of an input string and e is a constant called the degree of a given mcfg head grammars hg s introduced by pollard and tree adjoining grammars tag s introduced by joshi et al 1975 are also grammatical formalisms to describe the syntax of natural languages the paper also presents the following results on the generative capacities of hg s tag s and 2 mcfg s which are a subclass of mcfg s 1 the class hl of languages generated by hg s is the same as the one generated by tag s 2 hl is the same as the one generated by left wrapping hg s or right wrapping hg s which is a proper subclass of hg s 3 hl is properly included in the one generated by 2 mcfg s as a corollary of 1 it is also shown that hl is a substitution closed full afl

self avoiding walks with curvature energy on fractals self avoiding walks with a curvature dependent energy are studied with renormalization group methods on some fractal lattices fixed points corresponding to universal and non universal behaviours are generally present however initial conditions of the renormalization group recursions can prevent non universality when universality holds the persistence length is found to diverge much faster than in the periodic lattice as the curvature energy increases

a computer simulation study of anomalous diffusion on percolating clusters near to the critical point abstract we have verified the scaling hypothesis for the following quantities via a computer simulation the average probability density p r t p r 2 t d f 2 f z the diffusion coefficient d r t p t 2 d w 1 h t d 0 z and the mean square displacement msd r 2 t t 2 d w f 2 t where z r2 r2 t it was found that there exists a cross over value zc which divides interval z in two regions where the diffusion coefficient exhibits a power law dependence on variable z with the different exponents in the region z zc on the basis of the analogy of the average probability in the asymptotic form and the mean square displacement in the case of the self avoiding and ordinary random walk models the relation dw dw 1 was derived

molecular trajectory algorithm for random walks on percolation systems at criticality in two and three dimensions abstract random walk simulations based on a molecular trajectory algorithm are performed on critical percolation clusters the analysis of corrections to scaling is carried out it has been found that the fractal dimension of the random walk on the incipient infinite cluster is d w 2 873 0 008 in two dimensions and 3 78 0 02 in three dimensions if instead the diffusion is averaged over all clusters at the threshold not subject to the infinite restriction the corresponding critical exponent k is found to be k 0 3307 0 0014 for two dimensional space and 0 199 0 002 for three dimensional space moreover in our simulations the asymptotic behaviors of local critical exponents are reached much earlier than in other numerical methods

diffusion and fracton dimensionality on fractals and on percolation clusters the authors present simulations of diffusion on an exact fractal and on percolation clusters at criticality for two and three dimensions the results for the fractal support the rammal and toulouse proposition 1983 that ds n dn varies as b n s n the results for percolation are in excellent agreement with the alexander and orbach conjecture 1982 that the fracton dimensionality d 4 3

a grammatical approach to rna rna interaction prediction much attention has been paid to rna rna interaction involved in posttranscriptional regulation of gene expression although there have been a few studies on secondary structure prediction of interacting rnas using dynamic programming dp algorithms no grammar based approach has been proposed this paper provides a new modeling for rna rna interaction based on multiple context free grammar mcfg we present a polynomial time parsing prediction algorithm of the stochastic version of mcfg experimental results show that our approach is comparable to an existing work based on dp the mcfg based approach is more flexible than other dp based methods

random walk statistics on fractal structures we consider some statistical properties of simple random walks on fractal structures viewed as networks of sites and bonds range renewal theory mean first passage time etc asymptotic behaviors are shown to be controlled by the fractal d and spectral d dimensionalities of the considered structure a simple decimation procedure giving the value of d is outlined and illustrated in the case of the sierpinski gaskets recent results for the trapping problem the self avoiding walk and the true self avoiding walk are briefly reviewed new numerical results for diffusion on percolation clusters are also presented

new universality class for the permeability problem in a percolation cluster abstract the maximum flow that a percolation cluster can sustain with a gaussian distribution of individual bond capacities is calculated we show the existence of a new universality class if the probability distribution of bond capacities has a finite value in the limit of zero capacity

the relation between diffusion along fractal surfaces and observable diffusion fractal solid surfaces are presented as being built up from repeating fractal units of size xi equal to the surface upper self similarity cutoff taking into account the difference between real diffusion pathways of particles of size lambda and their observable projections the authors distinguish three types of observable diffusion i the classical law for lambda xi ii anomalous time dependence for lambda xi and short times iii normal time dependence but anomalous behaviour of the observable diffusion coefficient for lambda xi and long times crossovers occur from iii to i with increasing lambda and from ii to iii with increasing time

conductivity exponent in three dimensional percolation by diffusion based on molecular trajectory algorithm and blind ant rules abstract diffusion on random systems above and at their percolation threshold in three dimensions is carried out by a molecular trajectory method and a simple lattice random walk method respectively the classical regimes of diffusion on percolation near the threshold are observed in our simulations by both methods our monte carlo simulations by the simple lattice random walk method give the conductivity exponent 2 32 0 02 for diffusion on the incipient infinite clusters and 2 21 0 03 for diffusion on a percolating lattice above the threshold however while diffusion is performed by the molecular trajectory algorithm either on the incipient infinite clusters or on a percolating lattice above the threshold the result is found to be 2 26 0 02 in addition it takes less time step for diffusion based on the molecular trajectory algorithm to reach the asymptotic limit comparing with the simple lattice random walk

the transport exponent in percolation models with additional loops abstract several percolation models with additional loops were studied the transport exponents for these models were estimated numerically by means of a transfer matrix approach it was found that the transport exponent has a drastically changed value for some of the models this result supports some previous numerical studies on the vibrational properties of similar models with additional loops

learning from examples in large neural networks a statistical mechanical theory of learning from examples in layered networks at finite temperature is studied when the training error is a smooth function of continuously varying weights the generalization error falls off asymptotically as the inverse number of examples by analytical and numerical studies of single layer perceptrons we show that when the weights are discrete the generalization error can exhibit a discontinuous transition to perfect generalization for intermediate sizes of the example set the state of perfect generalization coexists with a metastable spin glass state

the notion of a strange fractal phenomenon disorder induced walk it is argued that a random system coupled with a walk may cause an unusual phenomenon which shares some of the opposite properties of fractals

homogeneity and spectral dimension of aggregation fractals the fracton or spectral dimension ds of two sutherland type ghost aggregation fractals is obtained for high space dimension by a scaling argument if as seems probable these fractals are homogeneous the results ds 2ln2 ln3 and ds 4 square root 17 1 support recent conjectures that ds is close to 4 3 for all homogeneous fractals however they are not consistent with the stronger conjecture that ds is exactly 4 3 for all such fractals