faster calculation of the percolation correlation length on spatial networks the divergence of the correlation length at criticality is an important phenomenon of percolation in two dimensional systems substantial speed ups to the calculation of the percolation threshold and component distribution have been achieved by utilizing disjoint sets but existing algorithms of this sort cannot measure the correlation length here we utilize the parallel axis theorem to track the correlation length as nodes are added to the system allowing us to utilize disjoint sets to measure for the entire percolation process with arbitrary precision in a single sweep this algorithm enables direct measurement of the correlation length in lattices as well as spatial network topologies and provides an important tool for understanding critical phenomena in spatial systems

statistics of red sites on elastic and full backbone abstract we investigated the number of red sites on the elastic and real backbone when right at the percolation threshold a spanning cluster exists between two sites at opposite faces of the lattice and found that it scales in the same way as in the case of percolation between two plates we also found that the number of common red sites scale similarly for both kinds of backbones for percolation between pairs of sites on opposite faces of the lattice our statistics for several quantities show that the exponent for the elastic backbone approaches the one for the full backbone as dimensionality is increased

greedy algorithms in disordered systems abstract we discuss search minimal path and minimal spanning tree algorithms and their applications to disordered systems greedy algorithms solve these problems exactly and are related to extremal dynamics in physics minimal cost path dijkstra and minimal cost spanning tree prim algorithms provide extremal dynamics for a polymer in a random medium the kpz universality class and invasion percolation without trapping respectively

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

reciprocal relations for the weight factors arising in the series expansion of backbone percolation functions the authors consider several properties of the backbone of percolation clusters the percolation model on a graph g is defined as follows each element vertex or edge of g is either open or closed the element e being open with probability pe independently of all other elements the u v backbone in any configuration is the subgraph consisting of all elements which belong to at least one open self avoiding path from a vertex u to a vertex v they show that the expected value of any u v backbone random variable z which depends only on the elements of the u v backbone e g z may be the number of edges or the length of the shortest path of the backbone is given by e z sigma k contained in p 1 mod k mod 1zk pi e in g k pe where p is the set of all self avoiding paths from u to v on g g k is the subgraph formed by all elements which belong to at least one self avoiding path of k and the weightzk is given by zk sigma j contained in k 1 mod jmod 1 zj where zj is the value of z when the backbone is g j certain reciprocal relations are found to hold between the weights and the backbone variables in some important cases using these results series expansions for the various properties can be obtained

heat diffusion maximal operators for laguerre semigroups with negative parameters abstract we prove l p boundedness for the maximal operator of the heat semigroup associated to the laguerre functions l k k when the parameter is greater than 1 namely the maximal operator is of strong type p p if p 1 and 2 2 p 2 when 1 0 if 0 there is strong type for 1 p the behavior at the end points is studied in detail

exact enumeration approach to fractal properties of the percolation backbone and 1 σ expansion an exact numeration approach is developed for the backbone fractal of the incipient infinite cluster at the percolation threshold the authors use this approach to calculate exactly the first low density expansion of lbb p for arbitrary system dimensionality d where lbb p is the mean of backbone bonds and p is the bond occupation probability standard series extrapolation methods provide estimates of the fractal dimension of the backbone for all d these disagree with the sierpinski gasket model of the backbone they also calculate the first low density expansions of lmin p and lred p which are respectively the mean number of bonds in the minimum path between i and j and the mean number of singly connected red bonds

epidemic models and percolation the authors argue that local epidemic models with immunisation are in the same universality class as percolation cluster growth models and show that the static exponents are equal to all orders in the epsilon expansion they calculate the dynamic exponent nu t 1 1 28 epsilon o epsilon 2 and relate this to other exponents involving the chemical distance

a geometrical interpretation of the chaotic state of inhomogeneous deterministic cellular automata abstract we propose a geometrical interpretation of the chaotic state of inhomogeneous cellular automata from the rules of the cellular automaton we construct a network the percolating phase of this network coincides with the chaotic phase of the cellular automaton we also report numerical tests of these ideas on several cellular automata

an algorithm for detecting percolating structures in periodic systems application to polymer networks we consider the problem of detecting a percolating structure in an off lattice model polymer system when periodic boundary conditions are used physically with increasing polymer density the point at which this first occurs is the gel point a connected structure spans all space and the system becomes solid like this problem is similar to that of finding connected paths in lattice bond percolation and site percolation models we show that algorithms that detect these structures in finite systems will not always yield the correct answer for a system that is periodically repeated in space uses periodic boundary conditions here we describe an algorithm that detects clusters that connect to themselves over an arbitrary number of periodic replicas because of the periodic replication this means that they are connected over all space for system sizes that are typically tractable in simulations we find a relatively minor proportion of configurations are mis classified for a lattice model however the fraction that is mis classified is significant for the polymer system mis classified configurations when included in the calculation of ensemble averages will include configurations with spurious physical properties our algorithm allows this to be corrected for

the bochner riesz means for fourier bessel expansions abstract the bochner riesz means for fourier bessel expansions are analyzed we prove a uniform two weight inequality with potential weights for these means the result provides necessary and sufficient conditions for boundedness moreover we obtain some corollaries regarding the convergence of these means and the boundedness of other operators related to the fourier bessel series

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

gradient method for thermal phase transitions the authors present a new numerical method of obtaining critical temperatures and exponents for ferromagnetic spin models and dynamical transitions this method is based on the spreading of damage in a temperature gradient they illustrate the method on the two dimensional ising model and present new results on the or ising spin glass in two dimensions they also obtain new critical exponents describing the behaviour of the width and the length of the damage front

the chemical distance distribution in percolation clusters the authors study the conditional probability density p r mod l for the geometrical distance r corresponding to a given chemical distance l for percolation clusters in two dimensions they argue that i p r mod l alxg exp axdelta where x r lnu g 2 5 or 0 3 delta 9 8 or 0 5 and nu 0 88 or 0 02 and ii there is a relation delta 1 nu 1 which is in good agreement with numerical data these results are derived by considering a special class of self avoiding walks consisting of chains which are chemical paths shortest paths on critical percolation clusters

percolation in cluster cluster aggregation processes numerical simulations of diffusion limited and reaction limited cluster cluster aggregation processes of identical particles are performed in a two dimensional box it is shown that for concentrations larger than a characteristic gel concentration the morphology of the resulting spanning cluster at the gel time t g exhibits a crossover length l c between percolation l l c and aggregation l c p the long range correlations are similar to that of percolation the vanishing links in the structure suggest that an homogeneous regime appears at small scales

invasion percolation new algorithms and universality classes employing highly efficient algorithms for simulating invasion percolation ip whose execution time scales as o mlog m or better for a cluster of m sites and for determining the backbone of the cluster we obtain precise estimates for the fractal dimensions of the sample spanning cluster the backbone and the minimal path in order to identify the universality classes of four different ip processes site and bond ip with and without trapping in two dimensions ip is characterized by two universality classes one each for ip without trapping and site and bond ip with trapping in a three dimensional site ip with and without trapping is in the universality class of random percolation while bond ip with trapping is in a distinct universality class which may be the same as that of optimal paths in strongly disordered media

red bond exponents of the critical and the tricritical ising model in three dimensions using the wolff and geometric cluster algorithms and finite size scaling analysis we investigate the critical ising and the tricritical blume capel models with nearest neighbor interactions on the simple cubic lattice the sampling procedure involves the decomposition of the ising configuration into geometric clusters each of which consists of a set of nearest neighboring spins of the same sign connected with bond probability p these clusters include the well known kasteleyn fortuin clusters as a special case for p 1 exp 2k where k is the ising spin spin coupling along the critical line k kc the size distribution of geometric clusters is investigated as a function of p we observe that unlike in the case of two dimensional tricriticality the percolation threshold in both models lies at pc 1 exp 2kc further we determine the corresponding red bond exponents as yr 0 757 2 and 0 501 5 for the critical ising and the tricritical blume capel models respectively on this basis we conjecture yr 1 2 for the latter model

jacobi transplantation revisited a transplantation theorem for jacobi series proved by muckenhoupt is reinvestigated by means of a suitable variant of calderon zygmund operator theory an essential novelty of our paper is weak type 1 1 estimate for the jacobi transplantation operator located in a fairly general weighted setting moreover l p estimates are proved for a class of weights that contains the class admitted in muckenhoupt s theorem

cumulant renormalisation group and its application to the incipient infinite cluster in percolation the authors introduce the cumulant method into the context of position space renormalisation group the advantage of this method over previous procedures is that one need not systematically evaluate all 2n states of each kadanoff cell where n is the number of elements sites or bonds in the cell they illustrate this method by calculating for bond percolation with an eight bond cell the critical exponents characterizing the following quantities i the mean number of backbone bonds ii the mean number of bonds in the minimum and maximum paths along the backbone iii the mean number of singly connected red bonds and iv the mean number of bonds in all the self avoiding walks connecting the two extreme points they derive previously obtained results of the conductivity and resistivity

the empirical mode decomposition and the hilbert spectrum for nonlinear and non stationary time series analysis a new method for analysing nonlinear and non stationary data has been developed the key part of the method is the empirical mode decomposition method with which any complicated data set can be dec

wetting velocity near the directed percolation threshold near the directed percolation threshold pd the wetting velocity upsilon scales as 1 upsilon approximately pd p theta the authors extrapolate from calculations on strips of finite width to show that on the two dimensional square lattice theta 1 73 or 0 02 this agrees with a scaling argument by barma and ray phys rev b vol 34 p 3403 1986 which predicts theta nu sub

the starry sky model of the percolation phase transition iii a new equilibrium equation and two kinds of the special elastic nodes abstract the equilibrium lame equation in two dimensional and three dimensional spaces can be presented as conductivity in magnetic field equation with lorenz force expression this allows us to apply the perturbation theory and generalize the lame equation and obtain a new equilibrium equation with all responses of compression and tension it means that compression in one direction leads to tension in the perpendicular direction and vice versa if the boundaries are fixed the solution in recurrent integral form for contributions to all orders of poisson ratio is then obtained zero order solution of lame equilibrium equation expresses the poisson ratio as ve p pc k k 0 3 and creates the special elastic nodes with maximum local deformation the first order solution gives mean values of effective lateral displacements or forces when lateral boundaries are fixed and expresses the bulk modulus as ke p pc 2t g where g 0 6 is a hall coefficient critical exponent it creates the special active elastic nodes where the whole sample microrotates around a new universality class of elasticity problem belongs to the hall coefficient universality class therefore the elasticity backbone coincides with the hall coefficient backbone with the fractal dimension d f b 2 25

a new percolation statistic with unusual properties in bond percolation on lattice with p pc there are unoccupied bonds which if occupied would immediately join the backbone in general when such a bond is occupied several tag end bonds may join the backbone as well the author defines ni to be the number of bonds that would join the backbone if such a bond i were occupied and n to be the average of ni taken over all such bonds on the lattice the author derives a formula for n and finds that for all lattices near pc n gamma bpc p pc 1 where gamma b is the exponent for the divergence of the backbone fraction identical formulae apply to site percolation

evidence for non universal exponents in bootstrap percolation bootstrap percolation bp models are systems where sites are initially randomly occupied those sites that do not maintain a suitable local environment of at least m occupied sites are successively removed current open questions concerning these systems now focus on the universality of critical exponents for the lower m values simulations indicate that for m 3 on the simple cubic lattice the exponent beta which characterizes the critical behaviour of the percolation probability is distinct from that of usual random percolation m 0 the authors quote pc 0 5717 or 0 0005 and beta 0 6 or 0 1 for m 3 they argue that the connectivity exponent nu for m 2 should be the same as that of usual random percolation and within the limits of numerical accuracy the authors observe that this appears to be the case for m 3 on the simple cubic lattice the implications of this surprising dichotomy are considered

eden growth on fractal media abstract simulations are presented for two dimensional eden growth into fractal networks we find that the fractal dimension of the eden cluster d e does not have a simple behavior when the cluster grows into the structure of a fractal network we also find that d e depends strongly on the minimal path between two arbitrary points in the network backbone and on the number of paths that appear between these two points at each scale

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

a sharp weighted transplantation theorem for laguerre function expansions abstract we find the sharp range of boundedness for transplantation operators associated with laguerre function expansions in l p spaces with power weights namely the operators interchanging l k and l k are bounded in l p y p if and only if 2 1 p 1 1 p 2 where min this improves a previous partial result by stempak and trebels which was only sharp for 0 our approach is based on new multiplier estimates for hermite expansions weighted inequalities for local singular integrals and a careful analysis of kanjin s original proof of the unweighted case as a consequence we obtain new results on multipliers riesz transforms and g functions for laguerre expansions in l p y p

backbone and elastic backbone of percolation clusters obtained by the new method of burning we introduce a method of obtaining information about percolation clusters in which the clusters are burned in different ways this method yields the backbone and the elastic backbone their number of loops shortest path and other quantities it works in any dimension and can be applied also to other clusters we obtain accurate estimates for the backbone exponents in two and three dimensions

geometrical properties of swendsen wang clusters abstract several geometrical properties of swendsen wang clusters are studied numerically for the 2d ising and 3 state potts models at the critical temperature the fractal dimensions of the total mass backbone elastic backbone shortest path and red bonds are determined the results agree with some theoretical predictions

backbone exponents of the two dimensional q state potts model a monte carlo investigation we determine the backbone exponent x b of several critical and tricritical q state potts models in two dimensions the critical systems include the bond percolation the ising the q 2 sqrt 3 3 and 4 state potts and the baxter wu model and the tricritical ones include the q 1 potts model and the blume capel model for this purpose we formulate several efficient monte carlo methods and sample the probability p2 of a pair of points connected via at least two independent paths finite size scaling analysis of p2 yields x b as 0 3566 2 0 2696 3 0 2105 3 and 0 127 4 for the critical q 2 sqrt 3 1 2 3 and 4 state potts model respectively at tricriticality we obtain x b 0 0520 3 and 0 0753 6 for the q 1 and 2 potts model respectively for the critical q 0 potts model it is derived that x b 3 4 from a scaling argument we find that at tricriticality x b reduces to the magnetic exponent as confirmed by the numerical results

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

topological loop intermittency in the three dimensional turbulence a new topological or loop type of intermittency is studied in three dimensional turbulence the spectral or fracton dimension ds 7 4 7 1 95 for this turbulence i e all states are localized this intermittency is different both from kolmogorov type unlocalized since ds 2 and from alexander orbach type strong localized helical fractons with ds 43 the corresponding energy spectral exponent is obtained to be equal to 1 7 1 65 these results are in good agreement with recent numerical spectral dimension and laboratory turbulent energy spectrum data the anomaly in the experimentally observed fractal dimension of clouds surface d 2 35 can be also explained in the framework of this intermittency phenomenon the results have been obtained by a renormalization of the noise like 1k energy spectrum in isotropic incompressible turbulence with taking into account of spontaneous parity breaking by virtual pairs of helical excitations

harmonic analysis operators associated with multidimensional bessel operators we establish that the maximal operator and the littlewood paley g function associated with the heat semigroup defined by multidimensional bessel operators are of weak type 1 1 we also prove that riesz transforms in the multidimensional bessel setting are of strong type p p for every 1 p

notes on the interpolation of discrete periodic signals using sinc function related approaches considering the sinc interpolation of discrete periodic signals this correspondence establishes the equivalence of two recently published methods furthermore the interpolation methods are equivalent to a fractional delay filter designed using the frequency sampling technique a numerically stable and computationally attractive form of the equivalent methods is presented

solutions of weinstein equations representable by bessel poisson integrals of bmo functions abstract we consider the weinstein type equation l u 0 on 0 0 where l t 2 x 2 1 x 2 with 1 in this paper we characterize the solutions of l u 0 on 0 0 representable by bessel poisson integrals of bmo functions as the ones satisfying certain carleson properties

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

on fourier interpolation error for band limited signals fourier interpolation is a useful tool to reconstruct continuous signals from discrete samples this correspondence derives fourier interpolation expression from fourier series for periodic signals and investigates its error expression and upper bound for general band limited signals two numerical examples are given to interpret the meaning of the formulas

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

sinc interpolation of discrete periodic signals the paper introduces a method for the sinc interpolation of discrete periodic signals the convolution of the sinc kernel with the infinite sequence of a periodic function is rewritten as a finite summation the method is equivalent to trigonometrical interpolation by fourier series expansion

non linear conductivity of granular superconductors a novel breakdown problem the authors study a percolating network of highly hysteretic josephson junctions in which each junction is modelled as a piecewise ohmic hysteretic device at the percolation threshold the average voltage drop across a network on a large l l grid is assumed to fall to zero as v approximately i ic l eta as the applied current is reduced to the critical current ic l the finite size monte carlo study gives the value eta 2 0 or 0 1 this value is remarkably close to the result eta 2 obtained by a scaling analysis of a continuum version of the model

bellman functions and dimension free lp estimates for the riesz transforms in bessel settings abstract in this article we prove dimension free l p boundedness of riesz transforms associated with a bessel differential operator we obtain explicit estimates of the l p norms for the bessel riesz transforms in terms of p establishing a linear behavior with respect to p we use the bellman function technique to prove a bilinear dimension free inequality involving poisson semigroups defined through this bessel operator

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

family of growth fractals with continuously tunable chemical dimension we introduce a new class of statistical growth fractals which is of interest because the chemical dimension d is continuously tunable we also study other exponents charac terising these fractals there has been considerable recent interest in uncovering the fashion in which the familiar laws of physics are modified for fractal objects in part because of the large number of important realisations of fractals in nature l 81 it has only recently become appreciated that the physics of fractals is determined by more than just the fractal dimension df which describes how the cluster mass n scales with the cluster radius r

static critical exponents from the dynamics of damage spreading and overhangs in the ising model with temperature gradient by studying damage spreading in spin models in a temperature gradient it is possible to determine not only the critical temperature and the correlation length exponent v but also a second critical exponent beta with high precision this is done by studying the dynamics of damage spreading in particular the overhangs which develop in the damage front the authors demonstrate the method for the two dimensional ising model using heat bath dynamics and discuss differences with metropolis dynamics

riesz transforms related to bessel operators riesz transforms r mu related to the bessel operators delta mu x mu 1 2 dx 2 mu 1 dx mu 1 2 are studied in this work we develop for r mu a theory that runs parallel to that for the euclidean hilbert transform it is proved that r mu is actually a calderon zygmund singular integral operator also r mu is seen to be the boundary value of the appropriate harmonic extension for this context finally we analyse weighted inequalities involving r mu

the data driven null models for information dissemination tree in social networks abstract for the purpose of detecting relatedness and co occurrence between users as well as the distribution features of nodes in spreading path of a social network this paper explores topological characteristics of information dissemination trees idt that can be employed indirectly to probe the information dissemination laws within social networks hence three different null models of idt are presented in this article including the statistical constrained 0 order idt null model the random rewire broken edge 0 order idt null model and the random rewire broken edge 2 order idt null model these null models firstly generate the corresponding randomized copy of an actual idt then the extended significance profile which is developed by adding the cascade ratio of information dissemination path is exploited not only to evaluate degree correlation of two nodes associated with an edge but also to assess the cascade ratio of different length of information dissemination paths the experimental correspondences of the empirical analysis for several sinaweibo idts and twitter idts indicate that the idt null models presented in this paper perform well in terms of degree correlation of nodes and dissemination path cascade ratio which can be better to reveal the features of information dissemination and to fit the situation of real social networks

shortest paths in percolation the separation of two points on a percolation network is characterised not only by the distance between them but also by the length of a path on the network which connects them the wetting velocity nu provides a measure of the lengths of the shortest connecting paths on the network above the percolation concentration pc along the easy axes nu is expected to vanish as p pc theta near pc while 1 nu is expected to vary as pd p theta near the directed percolation concentration pd the exponent theta is related to an exponent nu which characterises the shortest paths precisely at pc and which has recently been determined numerically the wetting velocity is calculated analytically on a randomly diluted bethe lattice and the values theta 1 2 and theta 1 with logarithmic corrections are found the direction dependence of nu is also investigated

computer simulations of randomly branching polymers annealed versus quenched branching structures we present computer simulations of three systems of randomly branching polymers in d 3 dimensions ideal trees and self avoiding trees with annealed and quenched connectivities in all cases we performed a detailed analysis of trees connectivities spatial conformations and statistical properties of linear paths on trees and compare the results to the corresponding predictions of flory theory we confirm that overall the theory predicts correctly that trees with quenched ideal connectivity exhibit less overall swelling in good solvent than corresponding trees with annealed connectivity even though they are more strongly stretched on the path level at the same time we emphasize the inadequacy of the flory theory in predicting the behaviour of other and equally relevant observables like contact probabilities between tree nodes we show then that contact probabilities can be aptly characterized by introducing a novel critical exponent which accounts for how they decay as a function of the node to node path distance on the tree

a precise determination of the backbone fractal dimension on two dimensional percolation clusters the backbone fractal dimension dbf is calculated on two dimensional percolation clusters at the percolation threshold studies are carried out on backbones defined in three different ways bus bar point to point and with fully periodic boundaries the estimates of dbf are obtained by measuring the variation of mass with radius for all clusters and by means of finite size scaling on the bus bar backbones all cases imply a value of dbf 1 64 or 0 01 for the backbone fractal dimension because of the high degree of self consistency of all the results the authors believe that this estimate represents a considerably improved accuracy

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

series expansion analysis of the backbone properties of two dimensional percolation clusters low density series expansions for the backbone properties of two dimensional bond percolation clusters are derived and analysed expansions for most of the 14 properties considered are new and are obtained to order on the square lattice and order on the triangular lattice earlier series work was confined to three properties of the square lattice and was to order the fractal dimension of the bonds or sites in the backbone is estimated to be and is intermediate between a previously conjectured field theory value and the latest monte carlo results the union intersection and length of the longest self avoiding paths are found to have the same fractal dimension which is close to and consistent with the field theory conjecture for on the other hand the union intersection and length of the shortest paths are found to have different dimensions and in the case of the intersection the triangular and square lattices are found to have sigificantly different dimensions the fractal dimension of the shortest path also appears to be non universal and we find for the square lattice and for the triangular lattice critical amplitude ratios are considered and found to be in agreement with theoretical inequalities

effect of loops on the vibrational spectrum of percolation network abstract we study the effects of adding loops to a critical percolation cluster on the diffusional and equivalently scalar elastic properties of the fractal network from the numerical calculations of the eigenspectrum of the transition probability matrix we find that the spectral dimension ds and the walk dimension dw change suddenly as soon as the floppy ends of a critical percolation cluster are connected together to form relatively large loops and we find that the additional inclusion of successively smaller loops only change these exponents little if at all this suggests that there is a new universality class associated with the loop enhanced percolation problem

current distribution on a three dimensional bond diluted random resistor network at the percolation threshold current and logarithm current distributions on a three dimensional random bond percolation cubic network were studied at the percolation threshold by computer simulations predictions of a hierarchical model that combine fractal structure and randomness agree with our numerical simulations in the thermodynamic limit the logarithm current distribution exhibits ann ln i i1 3 dependence below some characteristic currentic this distribution may scale with lni lnl but the data are insufficient to make this a definite conclusion due to the small range of lnl considered a study of the moments does not reveal this behavior and a study of the distribution itself is required

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

scaling relation for the critical exponents of the backbone of percolation clusters the author suggests that beta b the critical exponent of the backbone of percolation clusters is given by beta b 1 2 gamma 5 beta 1 where gamma and beta are susceptibility and order parameter exponents of percolation respectively the proposed relation agrees with the epsilon expansion of beta b given by harris and lubensky and provides accurate estimates of beta b at all d 1 this relation may suggest that the backbone exponents are not independent of other percolation exponents the author also suggests a more general relation for the order parameter exponent of m connected clusters

distribution of dangling ends on the incipient percolation cluster we study numerically and by scaling arguments the probabilityp m dm that a given dangling end of the incipient percolation cluster has a mass between m and m dm we nd by scaling arguments that p m decays with a power law p m m 1 with an exponent d b f df where df and d b f are the fractal dimensions of the cluster and its backbone respectively our numerical results yield 0 83 in d 2 and 0 74 in d 3 in very good agreement with theory c 1999 published by elsevier science b v all rights reserved

regular random fractals and the n parameter model two models of fractal objects are presented the first model represents a new class of fractals which is intermediate to statistical fractals and exact fractals in that each configuration of the random ensemble has an exact and adjustable fractal dimension the second model has a continuously adjustable backbone dimensions and can be used to directly generate conducting backbones

fourier acceleration of iterative processes in disordered systems technical details are given on how to use fourier acceleration with iterative processes such as relaxation and conjugate gradient methods these methods are often used to solve large linear systems of equations but become hopelessly slow very rapidly as the size of the set of equations to be solved increases fourier acceleration is a method designed to alleviate these problems and result in a very fast algorithm the method is explained for the jacobi relaxation and conjugate gradient methods and is applied to two models the random resistor network and the random central force network in the first model acceleration works very well in the second little is gained we discuss reasons for this we also include a discussion of stopping criteria

geometrical properties of single connected bonds in percolation clusters numerical simulation of the two dimensional bond percolation at the percolation threshold on an l l lattice shows that the radius of gyration of singly connected bonds scb scales as l1 this result confirms an assumption used to derive a lower bound on the critical exponent of elasticity a persistent anisotropy in the distribution of scb is found and used to estimate the ratio between the bulk and shear moduli of a random percolating system local anisotropy in the direction of scb disappears for large l

topological properties of diffusion limited aggregation and cluster cluster aggregation the detailed topological or connectivity properties of the clusters formed in diffusion limited aggregation dla and cluster cluster aggregation cca are considered for spatial dimensions d 2 3 and 4 specifically for both aggregation phenomena the authors calculate the fractal dimension dmin nu 1 defined by l approximately rd min where l is the shortest path between two points separated by a pythagorean distance r for cca they find that dmin increases monotonically with d presumably tending toward a limiting value dmin 2 at the upper critical dimensionality dc as found previously for lattice animals and percolation for dla on the other hand they find that dmin 1 within the accuracy of the calculations for d 2 3 and 4 suggesting the absence of an upper critical dimension they also discuss some of the subtle features encountered in calculating dmin for dla

cluster structure near the percolation threshold derives exact relations that allow one to describe unambiguously and quantitatively the structure of clusters near the percolation threshold pc in particular the author proves the relations p dpij dp lambda ij where p is the bond density pij is the pair connectedness function and lambda ij is the average number of cutting bonds between i and j from this relation it follows that the average number of cutting bonds between two points separated by a distance of the order of the connectedness length xi diverges as mod p pc mod 1 the remaining multiply connected bonds in the percolating backbone which lump together in blobs diverge with a dimensionality dependent exponent he also shows that in the cell renormalisation group of reynolds et al 1978 1980 the thermal eigenvalue is simply related to the average number of cutting bonds in the spanning cluster he discusses a percolation model in which the blobs can be controlled by varying a parameter and study the influence on the critical exponents

fractal properties of the percolating backbone in three dimensions a recursive algorithm to determine the backbone of the two dimensional percolating cluster is adjusted to measure the fractal properties of the backbone of the three dimensional percolating cluster the specific properties measured are the fractal dimensionality of the backbone dbbf 1 75 or 0 04 the chemical dimensionality dt 1 26 or 0 03 and the minimum path dimensionality dmin 1 39 or 0 03

a new algorithm to extract the backbone in a random resistor network the authors present an efficient algorithm to determine the backbone the current carrying part of a lattice in the random resistor network problem it reduces mainly to the determination of the cluster structure on two lattices its efficiency is due to the use of the dual lattice and therefore is specific to the two dimensional case

shortest path fractal dimension for percolation in two and three dimensions we carry out a high precision monte carlo study of the shortest path fractal dimension d mathrm min for percolation in two and three dimensions using the leath alexandrowicz method which grows a cluster from an active seed site a variety of quantities are sampled as a function of the chemical distance including the number of activated sites a measure of the radius and the survival probability by finite size scaling we determine d mathrm min 1 130 phantom rule 0 16em 0ex 77 2 and 1 3756 6 in two and three dimensions respectively the result in two dimensions rules out the recently conjectured value d mathrm min 217 192 deng et al phys rev e 81 020102 r 2010

order propagation near the percolation threshold a new monte carlo method for studying bond percolation clusters is developed and used to identify new critical quantities associated with the percolation threshold the bonds in each cluster are partitioned into three distinct connectivity classes red singly connected backbone bonds blue multiply connected backbone bonds and yellow non backbone bonds often called dangling ends among the new cluster properties studied are the mean number of red bonds a critical quantity diverging at p c with exponent y r approximately 1 and the length of the shortest connected path through the cluster which is critical with exponent y min 1 35 0 02 for all cluster properties studies the authors also compute averages over only the largest clusters the corresponding critical exponents are found to be significantly different from those obtained by averaging over clusters of all sizes

comments on sinc interpolation of discrete periodic signals in a recent paper by t schanze see ibid vol 43 p 1502 3 1995 the convolution of the sinc kernel with the infinite sequence of a periodic function was expressed as a finite summation the expression obtained however is not numerically stable when evaluated at or near integer values of time this correspondence presents a numerically stable formulation that is equivalent to the results reported in the above cited paper and that when sampled is also shown to be equivalent to the inverse discrete fourier transform idft

functional calculus for the laguerre operator in this paper we study the boundedness of spectral multipliers associated to the multidimensional laguerre operator open image in new window it is well known that for special values of the analysis of the laguerre operator can be interpreted as the analysis of the ornstein uhlenbeck operator acting on polyradial functions exploiting this relation we prove that if m is a bounded holomorphic function in the sector open image in new window satisfying suitable hormander type conditions on the boundary then the spectral operator m open image in new window is bounded on lp with respect to the laguerre measure we also prove that holomorphy in the sector s is a necessary condition for multipliers whose norm is invariant under dilations

position space renormalisation group study of shortest paths in percolation clusters the average length of the shortest paths along the backbone of the incipient infinite cluster is expected to diverge near the percolation threshold pc as mod p pc mod nu s with nu s different from the correlation length exponent nu the author applies the position space renormalisation group psrg technique to obtain nu s both for square and cubic lattices and for a triangular lattice in two dimensions the results obtained nu s 1 49 for d 2 and 1 21 for d 3 are in very good agreement with the existing results obtained by different methods

dimension free lp estimates for single riesz transforms via an h joint functional calculus abstract by using an h joint functional calculus for strongly commuting operators we derive a scheme to deduce the l p boundedness of certain d dimensional riesz transforms from the l p boundedness of appropriate one dimensional riesz transforms moreover the l p bounds we obtain are independent of the dimension the scheme is applied to riesz transforms connected with orthogonal expansions and discrete riesz transforms on products of groups with polynomial growth

universal substructures of percolation clusters the skeleton the authors define the skeleton of a cluster aggregate as the set of all sites belonging to the shortest paths connecting a chosen site with the lth chemical shell surrounding that site the fractal properties of skeletons of percolation clusters at criticality have been studied and it is inferred that the mass of the skeleton ms scales with the chemical distance l for l l as ms approximately lls where dls 1 is universal for l or d or 6 numerical evidence which supports this conclusion is presented for d 2 and an analytical proof is given for d 6

damage spreading in a gradient we propose a new method of analyzing the frozen chaotic transition in a cellular automaton by propagating damage in a gradient we obtain estimations forpc and for the critical exponents for the kauffman model and the mixture of or and xor rules

maximal operators of exotic and non exotic laguerre and other semigroups associated with classical orthogonal expansions classical settings of discrete and continuous orthogonal expansions like those of laguerre bessel and jacobi are associated with second order differential operators playing the role of the laplacian these depend on certain type parameters that are usually restricted to a half line or a product of half lines if higher dimensions are considered following earlier research done by hajmirzaahmad we deal in this paper with laplacians in the above mentioned contexts with no restrictions on the type parameters and bring to attention naturally associated orthogonal systems that in fact involve the classical ones but are different this reveals new frameworks related to classical orthogonal expansions and thus new potentially rich research areas at least from the harmonic analysis perspective to support the last claim we focus on maximal operators of multi dimensional laguerre bessel and jacobi semigroups with unrestricted type parameters and prove that they satisfy weak type 1 1 estimates with respect to the appropriate measures generally these measures are not locally finite which makes a contrast with the classical situations and generates new difficulties a significant result of the paper is a new proof of the weak type 1 1 estimate for the classical multi dimensional laguerre semigroup maximal operator

field theoretic approaches to biconnectedness in percolating systems two field theoretic formulations for the percolation problem are presented from which the critical exponents describing the backbone of the infinite cluster at the percolation threshold are obtained at high spatial dimension d the order parameter exponent for the backbone beta 2 is given by beta 2 2 beta psi 2 nu where beta is the critical exponent for the density of the infinite cluster and psi 2 is a new crossover exponent in mean field theory psi 2 0 and for d 6 epsilon psi 2 2 epsilon 2 49 o epsilon 3 presumably psi 2 d is a smooth function of d for d d where numerical and theoretical work indicates that d is about 3 the result indicates that the fractal dimensionality of the backbone is given in terms of the percolation exponents as gamma nu psi 2

application of fractal concepts to polymer statistics and to anomalous transport in randomly porous media we consider the application of fractal concepts to polymer statistics and to anomalous transport in randomly porous media it is found that answers to interesting physics questions can be expressed in terms of several new fractal dimensions in addition to the fractal dimensiond f 1 d the fractal dimension of the backbone arises in connection with electric current flow 2 d red the fractal dimension of the singly connected bonds in the backbone arises in connection with its equivalence to the thermal scaling power 3 d e the fractal dimension of the of the elastic backbone 4 d u the fractal dimension of the unscreened perimeter arises in connection with the viscosity singularity at the gelation threshold 5 d min the fractal dimension of the minimum path or chemical distance between two sites arises in co nnection with the aharony stauffer conjecture 6 dw the fractal dimension of a random walk 7 d g the fractal dimension of growth sites that arise as a random walk creates a cluster relations among these fractal dimensions are discussed some of which can be proved and others of which are conjectures whose validity has been established only in certain limiting cases

the three dimensional ising model in a temperature gradient abstract the critical temperature t c and the correlation length exponent for the three dimensional ising model is determined via the gradient method by numerical studies of systems of size up to 360 3 we determine the correlation length exponent 0 62 0 01 in comparison to the best estimates ranging from 0 624 to 0 630

electrical conductivity of finite size percolation networks calculating by a monte carlo technique the conductivities of small two dimensional square bond percolation networks 4 4 to 50 50 at a limited number of probability values from the critical region to full conductivity the authors have been able to show by an original interpolation technique that in the critical region the conductivity function does indeed obey a universal fisher type finite size scaling function they further show that this technique permits them to deduce values of the critical exponents for both the conductivity and the correlation length even from calculations on relatively small networks