continuum percolation in two dimensions monte carlo tests of scaling and universality for non interacting discs detailed results are reported for the connectivity properties of a system of discs of unit radius free to be situated anywhere within a square of area 2l2 ordinary lattke percolation would correspond to the discs being situated on the vertices of a j2l x j2l lattice computer simulations are carried out for a sequence of increasing system sizes ranging from l 20 to l 1000 for each value of l a large number of realisations are generated for 25 values of the disc concentration x we calculate a variety of estimates for the threshold parameter x as well as the critical exponents p y t and v our exponent estimates are in close agreement with accepted values for ordinary lattice percolation therefore this continuum system appears to be in the same universality class as lattice percolation

exact critical exponent for the shortest path scaling function in percolation it is shown that the critical exponentg1 related to pair connectiveness and shortest path or chemical distance scaling recently studied by porto et al dokholyan et al and grassberger can be found exactly in two dimensions by using a crossing probability result of cardy with the outcomeg1d 25 24 this prediction is consistent with existing simulation results

critical properties of random site percolation in two and three dimensions a monte carlo study abstract we measure the critical exponents of two dimensional and three dimensional random site percolation and find excellent agreement with nienhuis exact results two dimensions and good agreement with other numerical work three dimensions we also measure the correlation length amplitude ratio and the mean cluster size amplitude ratio in two and three dimensions we report values of 4 0 0 5 2 d 2 0 0 5 3 d and 75 40 26 2 d and 8 3 2 3 d for the correlation length and the mean cluster size amplitude ratios respectively a direct measurement is made of the fractal dimension d f at the critical point p c 0 50 of the triangular lattice d f is 1 90 0 01 while at p c 0 3117 for the simple cubic lattice d f is 2 50 0 02

multifractal hurst analysis of crude oil prices abstract daily records of international crude oil prices are studied using multifractal analysis methods rescaled range hurst analysis provides evidence that the crude oil market is a persistent process with long run memory effects on the other hand height height correlation analysis reveals evidence of multifractal structures in the sense that the crude oil dynamics displays mixing of rough hurst exponents the existence of two characteristic time scales in the order of weeks and quarters is discovered and the corresponding prices dynamics are extracted using moving average based filtering these results seem to demonstrate that the crude oil market is consistent with the random walk assumption only at time scales of the order of days to weeks a plausible oil price formation mechanism is discussed in terms of the market dynamics at three different time scales

multifractal spectra of multi affine functions abstract self affine functions f x with multiscaling height correlations cq x xqhq are described in terms of the standard multifractal formalism with a modified assumption for the partition the corresponding quantities and expressions are shown to exhibit some characteristic differences from the standard ones according to our calculations the f type spectra are not uniquely determined by the hq spectrum but depend on the particular choice which is made for the dependence of n on x where n is the number of points over which the average is taken our results are expected to be relevant in the analysis of signal type data obtained in experiments on systems with an underlying multiplicative process

scaling of the distribution of shortest paths in percolation we present a scaling hypothesis for the distribution function of the shortest paths connecting any two points on a percolating cluster which accounts for i the effect of the finite size of the system and ii the dependence of this distribution on the site occupancy probability p we test the hypothesis for the case of two dimensional percolation

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

fractional calculus and continuous time finance in this paper we present a rather general phenomenological theory of tick by tick dynamics in financial markets many well known aspects such as the levy scaling form follow as particular cases of the theory the theory fully takes into account the non markovian and non local character of financial time series predictions on the long time behaviour of the waiting time probability density are presented finally a general scaling form is given based on the solution of the fractional diffusion equation

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

statistical properties of deterministic threshold elements the case of market price abstract we analyze statistical properties of a set of deterministic threshold elements which is introduced as a model for the stock market the macroscopic variable of the stock market price shows seemingly stochastic fluctuation with a f 2 power spectrum consistent with real economic fluctuations the maximum lyapunov exponent is estimated to be zero indicating that the system is at the edge of chaos

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

summability of solutions of the dirichlet problem for nonlinear elliptic equations with right hand side in logarithmic classes close to l1 abstract in this paper we consider the dirichlet problem for a class of nonlinear elliptic second order equations with right hand side in l 1 we describe the summability properties of entropy and weak solutions of this problem in the case where the right hand side belongs to classes close to l 1 and defined by the logarithmic function or its compositions of arbitrary multiplicity

a critical exponent for shortest path scaling in continuum percolation we carry out monte carlo experiments to study the scaling behavior of shortest path lengths in continuum percolation these studies suggest that the critical exponent governing this scaling is the same for both continuum and lattice percolation we use splitting a technique that has not yet been fully exploited in the physics literature to increase the speed of our simulations this technique can also be applied to other models where clusters are grown sequentially

on the minority game analytical and numerical studies abstract we investigate further several properties of the minority game we have recently introduced we explain the origin of the phase transition and give an analytical expression of 2 n in the n 2 m region the ability of the players to learn a given payoff is also analyzed and we show that the darwinian evolution process tends to a self organized state in particular the lifetime distribution is a power law with exponent 2 furthermore we study the influence of identical players on their gain and on the systems performance finally we show that large brains always take advantage of small brains

the scaling properties of stock markets based on modified multiscale multifractal detrended fluctuation analysis abstract here we propose the new method dh mma based on multiscale multifractal detrended fluctuation analysis mma to investigate the scaling properties in stock markets it is demonstrated that our approach can provide a more stable and faithful description of the scaling properties in comprehensive range rather than fixing the window length and slide length it allows the assessment of more universal and subtle scaling characteristics we illustrate dh mma by selecting power law artificial data sets and six stock markets from us and china the us stocks exhibit very strong multifractality for positive values of q however the chinese stocks show stronger multifractality for negative q than positive q in general the us stock markets show similar behaviors but chinese stock markets display distinguishing characteristics

non linear elliptic and parabolic equations involving measure data in this paper we prove the existence of solutions for equations of the type div a du f in a bounded open set u 0 on where a is a possibly non linear function satisfying some coerciveness and monotonicity assumptions and f is a bounded measure we also consider the equation div a du g u f in u 0 on with f l1 or f m u u 0 and the parabolic equivalent of the first elliptic equation

vanishing gaps in 1d bandstructures the criterion for vanishing or zero energy gaps in the spectrum of 1d electronic bandstructures is obtained from perturbation theory it is in agreement with examples of periodic potentials which never exhibit zero gaps as well as with those which may including certain smooth periodic potentials the results have relevance to recent work in the field of semiconductor layered heterostructures or superlattices the analysis should also have implications for other problems that involve hill s determinant

continuum percolation thresholds in two dimensions a wide variety of methods have been used to compute percolation thresholds in lattice percolation the most powerful of these methods consists of microcanonical simulations using the union find algorithm to efficiently determine the connected clusters and in two dimensions using exact values from conformal field theory for the probability at the phase transition that various kinds of wrapping clusters exist on the torus we apply this approach to percolation in continuum models finding overlaps between objects with real valued positions and orientations in particular we find precise values of the percolation transition for disks squares rotated squares and rotated sticks in two dimensions and confirm that these transitions behave as conformal field theory predicts the running time and memory use of our algorithm are essentially linear as a function of the number of objects at criticality

multilevel splitting for estimating rare event probabilities we analyze the performance of a splitting technique for the estimation of rare event probabilities by simulation a straightforward estimator of the probability of an event evaluates the proportion of simulated paths on which the event occurs if the event is rare even a large number of paths may produce little information about its probability using this approach the method we study reinforces promising paths at intermediate thresholds by splitting them into subpaths which then evolve independently if implemented appropriately this has the effect of dedicating a greater fraction of the computational effort to informative runs we analyze the method for a class of models in which roughly speaking the number of states through which each threshold can be crossed is bounded under additional assumptions we identify the optimal degree of splitting at each threshold as the rarity of the event increases it should be set so that the expected number of subpaths reaching each threshold remains roughly constant thus implemented the method is provably effective in a sense appropriate to rare event simulations these results follow from a branching process analysis of the method we illustrate our theoretical results with some numerical examples for queueing models

theory of self avoiding walks on percolation fractals a phenomenological approach which takes into account the basic geometry and topology of percolation fractal structures and of self avoiding walks saw is used to derive a new expression for the flory exponent describing the average radius of gyration of saws on fractals we focus on the radius of gyration and discuss the importance of the intrinsic fractal dimensions of percolation clusters in determining the lower and upper critical dimensions of saws the mean field version of our new formula corresponds to the aharony and harris expression who used the standard flory approach for its derivation

topological properties of percolation clusters percolation properties including the total cluster mass s the shell mass b and the linear geometrical size r are studied as a function of the topological chemical distance parameter l all critical exponents are shown to be related to an apparently new exponent nu defined by r approximately lv critical exponents are calculated exactly for percolation clusters on the cayley tree a model for 6d percolation for which s approximately l2 b approximately l and r2 approximately l for diffusion on such clusters one finds that l3 approximately t numerical estimates of the exponents are obtained for other dimensions a conjecture which relates nu to beta and nu is discussed

dynamic numerical models of stock market price from microscopic determinism to macroscopic randomness abstract a variant of threshold dynamics is introduced to model the behaviors of a large assembly of dealers in a stock market although the microscopic evolution dynamics is deterministic the collective behaviors such as market prices show seemingly stochastic fluctuations the statistical properties of market price change can be well approximated by a simple discrete langevin type equation with random amplification the macroscopic stochastic equation is solved both numerically and analytically showing that the market price change generally follow power law distributions in the steady state the reason for the appearance of rapid decay in the distribution tails are discussed

pruned enriched rosenbluth method simulations of θ polymers of chain length up to 1 000 000 we present an algorithm for simulating flexible chain polymers it combines the rosenbluth rosenbluth method with recursive enrichment although it can be applied also in more general situations it is most efficient for three dimensional theta polymers on the simple cubic lattice there it allows high statistics simulations of chains of length up to n 10 sup 6 for storage reasons this is feasable only for polymers in a finite volume for free theta polymers in infinite volume we present very high statistics runs with n 10000 these simulations fully agree with previous simulations made by hegger and grassberger j chem phys bold 102 6681 1995 with a similar but less efficient algorithm showing that logarithmic corrections to mean field behavior are much stronger than predicted by field theory but the finite volume simulations show that the density inside a collapsed globule scales with the distance from the theta point as predicted by mean field theory in contrast to claims in the work mentioned above in addition to the simple cubic lattice we also studied two versions of the bond fluctuation model but with much shorter chains finally we show that our method can be applied also to off lattice models and illustrate this with simulations of amore model studied in detail by freire ital et al macromolecules bold 19 452 1986 and later work copyright ital 1997 ital the american physical society less

power law distributions in empirical data power law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man made phenomena unfortunately the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution the part of the distribution representing large but rare events and by the difficulty of identifying the range over which power law behavior holds commonly used methods for analyzing power law data such as least squares fitting can produce substantially inaccurate estimates of parameters for power law distributions and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all here we present a principled statistical framework for discerning and quantifying power law behavior in empirical data our approach combines maximum likelihood fitting methods with goodness of fit tests based on the kolmogorov smirnov ks statistic and likelihood ratios we evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches we also apply the proposed methods to twenty four real world data sets from a range of different disciplines each of which has been conjectured to follow a power law distribution in some cases we find these conjectures to be consistent with the data while in others the power law is ruled out

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

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

pair connectedness and shortest path scaling in critical percolation we present high statistics data on the distribution of shortest path lengths between two near by points on the same cluster at the percolation threshold our data are based on a new and very efficient algorithm for d 2 they clearly disprove a recent conjecture by m porto et al 1998 phys rev e 58 r5205 our data also provide upper bounds on the probability that two near by points are on different infinite clusters

distribution of shortest paths in percolation we present a scaling ansatz for the distribution function of the shortest paths connecting any two points on a percolating cluster which accounts for i the eect of the nite size of the system and ii the dependence of this distribution on the site occupancy probability p we present evidence supporting the scaling ansatz for the case of two dimensional percolation c 1999 elsevier science b v all rights reserved

a parallel multigrid algorithm for percolation clusters a new parallel cluster finding algorithm is formulated by using multigrid relaxation methods very similar to those used for differential equation solvers for percolation clusters this approach drastically reduces critical slowing down relative to local or scan relaxation methods numerical studies of scaling properties with system size are presented in the case of the 2d percolation clusters of the swendsen wang ising dynamics running on the connection machine

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

phase transition of dynamical herd behaviors for yen dollar exchange rates abstract we study the herd behavior and the phase transition for the yen dollar exchange rate in the japanese financial market it is obtained that the probability distribution of returns satisfies the power law behavior p r r with scaling exponents 3 11 2 81 and 2 29 at time intervals 1 min 30 min and 1 h the crash region in which the probability density increases with the increasing return appears when the herding parameter h satisfies h 2 33 for the case of 30 min we especially obtain that no crash occurs 30 min and that the probability distribution of price returns occurs in the phase transition at 30 min

correlations in the bond future market we analyze the time series of overnight returns for the bund and btp futures exchanged at liffe london the overnight returns of both assets are mapped onto a one dimensional symbolic dynamics random walk the bond walk during the considered period october 1991 january 1994 the bund future market opened earlier than the btp future one the crosscorrelations between the two bond walks as well as estimates of the conditional probability show that they are not independent however each walk can be modeled by means of a trinomial probability distribution monte carlo simulations confirm that it is necessary to take into account the bivariate dependence in order to properly reproduce the statistical properties of the real world data various investment strategies have been devised to exploit the prior information obtained by the aforementioned analysis

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

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

postbreakthrough behavior in flow through porous media we numerically simulate the traveling time of a tracer in convective flow between two points injection and extraction separated by a distance r in a model of porous media d 2 percolation we calculate and analyze the traveling time probability density function for two values of the fraction of connecting bonds p the homogeneous case p 1 and the inhomogeneous critical threshold case p p c we analyze both constant current and constant pressure conditions at p p c the homogeneous p 1 case serves as a comparison base for the more complicated p p c situation we find several regions in the probability density of the traveling times for the homogeneous case p 1 and also for the critical case p p c for both constant pressure and constant current conditions for constant pressure the first region i p corresponds to the short times before the flow breakthrough occurs when the probability distribution is strictly zero the second region ii p corresponds to numerous fast flow lines reaching the extraction point with the probability distribution reaching its maximum the third region iii p corresponds to intermediate times and is characterized by a power law decay the fourth region iv p corresponds to very long traveling times and is characterized by a different power law decaying tail the power law characterizing region iv p is related to the multifractal properties of flow in percolation and an expression for its dependence on the system size l is presented the constant current behavior is different from the constant pressure behavior and can be related analytically to the constant pressure case we present theoretical arguments for the values of the exponents characterizing each region and crossover times our results are summarized in two scaling assumptions for the traveling time probability density one for constant pressure and one for constant current we also present the production curve associated with the probability of traveling times which is of interest to oil recovery

renyi s information transfer between financial time series a b s t r a c t in this paper we quantify the statistical coherence between financial time series by means of the renyi entropy with the help of campbell s coding theorem we show that the renyi entropy selectively emphasizes only certain sectors of the underlying empirical distribution while strongly suppressing others this accentuation is controlled with renyi s parameter q to tackle the issue of the information flow between time series we formulate the concept of renyi s transfer entropy as a measure of information that is transferred only between certain parts of underlying distributions this is particularly pertinent in financial time series where the knowledge of marginal events such as spikes or sudden jumps is of a crucial importance we apply the renyian information flow to stock market time series from 11 world stock indices as sampled at a daily rate in the time period 02 01 1990 31 12 2009 corresponding heat maps and net information flows are represented graphically a detailed discussion of the transfer entropy between the dax and s p500 indices based on minute tick data gathered in the period 02 04 2008 11 09 2009 is also provided our analysis shows that the bivariate information flow between world markets is strongly asymmetric with a distinct information surplus flowing from the asia pacific region to both european and us markets an important yet less dramatic excess of information also flows from europe to the us this is particularly clearly seen from a careful analysis of renyi information flow between the dax and s p500 indices

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

lacunarity and percolation phase transitions the lacunarity is computed for randomly diluted two dimensional lattices in terms of the chemical distance the crossover behaviour and the question of universality are discussed

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

on the alexandrowicz relation between static and dynamic ising exponents abstract the cluster growth equation known from nucleation theory is found to reproduce the alexandrowicz relation if the growth probability is assumed to be proportional to the volume per site of the critical cluster supposing weak q dependence of the fractal dimension of the chemical distance in critical clusters of the q state potts model the relation is roughly fulfilled

scaling in the market of futures the price time series of the italian government bonds btp futures is studied by means of scaling concepts originally developed for random walks in statistical physics the series of overnight price differences is mapped onto a one dimensional random walk the bond walk the analysis of the root mean square fluctuation function and of the auto correlation function indicates the absence of both short and long range correlations in the bond walk a simple monte carlo simulation of a random walk with trinomial probability distribution is able to reproduce the main features of the bond walk

multifractal diffusion entropy analysis optimal bin width of probability histograms abstract in the framework of multifractal diffusion entropy analysis we propose a method for choosing an optimal bin width in histograms generated from underlying probability distributions of interest the method presented uses techniques of renyi s entropy and the mean squared error analysis to discuss the conditions under which the error in the multifractal spectrum estimation is minimal we illustrate the utility of our approach by focusing on a scaling behavior of financial time series in particular we analyze the s p500 stock index as sampled at a daily rate in the time period 1950 2013 in order to demonstrate a strength of the method proposed we compare the multifractal spectrum for various bin widths and show the robustness of the method especially for large values of q for such values other methods in use e g those based on moment estimation tend to fail for heavy tailed data or data with long correlations connection between the spectrum and renyi s q parameter is also discussed and elucidated on a simple example of multiscale time series

a self adjusted monte carlo simulation as a model for financial markets with central regulation abstract properties of the self adjusted monte carlo algorithm applied to 2d ising ferromagnet are studied numerically the endogenous feedback form expressed in terms of the instant running averages is suggested in order to generate a biased random walk of the temperature that converges to criticality without an external tuning the robustness of a stationary regime with respect to partial accessibility of the information is demonstrated several statistical and scaling aspects have been identified which allow to establish an alternative spin lattice model of the financial market it turns out that our model alike model suggested by bornholdt int j mod phys c 12 2001 667 may be described by levy type stationary distribution of feedback variations with unique exponent 1 3 3 however the differences reflected by hurst exponents suggest that resemblances between the studied models seem to be non trivial

precise determination of the critical threshold and exponents in a three dimensional continuum percolation model we present a large scale computer simulation of the prototypical three dimensional continuum percolation model consisting of a distribution of overlapping spatially uncorrelated spheres by using simulations of up to particles and studying the finite size scaling of various effective percolation thresholds we obtain a value of this value is significantly smaller than the values obtained for simulations that have been carried out using smaller systems employing this value of and systems of size l 160 relative to a sphere of unit radius we also obtain estimates of the critical exponents and for the continuum system and show that the values are different than those obtained using previous values of

probability distribution of the shortest path on the percolation cluster its backbone and skeleton we consider the mean distribution functions ensuremath phi r mathcal l ensuremath phi mathrm b r mathcal l and ensuremath phi mathrm s r mathcal l giving the probability that two sites on the incipient percolation cluster on its backbone and on its skeleton respectively connected by a shortest path of length l are separated by an euclidean distance r following a scaling argument due to de gennes for self avoiding walks we derive analytical expressions for the exponents g 1 d f d mathrm min ensuremath d and g 1 mathrm b g 1 mathrm s 3d mathrm min ensuremath d which determine the scaling behavior of the distribution functions in the limit x ensuremath equiv r mathcal l stackrel ifmmode tilde else fi ensuremath nu ensuremath ll 1 i e ensuremath phi r mathcal l ensuremath propto mathcal l ensuremath stackrel ifmmode tilde else fi ensuremath nu d x g 1 ensuremath phi mathrm b r mathcal l ensuremath propto mathcal l ensuremath stackrel ifmmode tilde else fi ensuremath nu d x g 1 mathrm b and ensuremath phi mathrm s r mathcal l ensuremath propto mathcal l ensuremath stackrel ifmmode tilde else fi ensuremath nu d x g 1 mathrm s with stackrel ifmmode tilde else fi ensuremath nu ensuremath equiv 1 d mathrm min where d f and d mathrm min are the fractal dimensions of the percolation cluster and the shortest path respectively the theoretical predictions for g 1 g 1 mathrm b and g 1 mathrm s are in very good agreement with our numerical results

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

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

general conditions for limit summability of solutions of nonlinear elliptic equations with l1 data we consider the dirichlet problem for nonlinear elliptic second order equations in divergence form with an l1 right hand side f by the function f we define other function with the values which are equal to the integrals of the function f over the sets f s s 0 under rather general integral conditions on the latter function we prove that the entropy solution of the problem under consideration belongs to some limit lebesgue and sobolev spaces

transmission of information and herd behavior an application to financial markets we propose a model for stochastic formation of opinion clusters modeled by an evolving network and herd behavior to account for the observed fat tail distribution in returns of financial price data the only parameter of the model is h the rate of information dispersion per trade which is a measure of herding behavior for h below a critical h the system displays a power law distribution of the returns with exponential cutoff however for hg h an increase in the probability of large returns is found and may be associated with the occurrence of large crashes

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

surface and edge exponents for the spreading of 3d percolation monte carlo results are presented for the spreading of 3d percolation where the seed consists of an infinite straight line this line can either be in the bulk of the material into which spreading is possible on a planar surface of the material or on a rectangular edge in the last two cases spreading occurs only into angular regions with 180 degrees resp 90 degrees while the mean distance grows at p pc in all three cases with the same exponent the average number of growth sites grows resp vanishes in all three cases with different exponents in particular this implies that at p pc the sites on the surface of a big cube belonging to the maximal cluster within that cube have fractal dimension 1

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

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

multifractality of stock markets based on cumulative distribution function and multiscale multifractal analysis abstract considering the diverse application of multifractal techniques in natural scientific disciplines this work underscores the versatility of multiscale multifractal detrended fluctuation analysis mma method to investigate artificial and real world data sets the modified mma method based on cumulative distribution function is proposed with the objective of quantifying the scaling exponent and multifractality of nonstationary time series it is demonstrated that our approach can provide a more stable and faithful description of multifractal properties in comprehensive range rather than fixing the window length and slide length our analyzes based on cdf mma method reveal significant differences in the multifractal characteristics in the temporal dynamics between us and chinese stock markets suggesting that these two stock markets might be regulated by very different mechanism the cdf mma method is important for evidencing the stable and fine structure of multiscale and multifractal scaling behaviors and can be useful to deepen and broaden our understanding of scaling exponents and multifractal characteristics

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

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

the tortuosity of occupied crossings of a box in critical percolation we consider the length of an occupied crossing of a box of size 0 n 0 3n d 1 in the short direction in standard bernoulli bond percolation on d at criticality let s n be the length of the shortest such crossing it is believed that s n 1 c in some sense for somec 0 here we show that if the correlation length p satisfies p c 1 n 1 logn 1 the assumption p c 3 p c p with 0 we also obtain a result about the length of optimal paths in first passage percolation

multifractal features of financial markets abstract we study the tick dynamical behavior of three assets in financial markets the kospi the won dollar and yen dollar exchange rates using the rescaled range r s analysis the multifractal hurst exponents with long run memory effect can be obtained from those assets and we discuss whether there exists the crossover or not for the hurst exponents at characteristic time scales particularly we find that the probability distribution of returns approaches to a lorentz distribution different from gaussian properties

binomial markov switching multifractal model with skewed t innovations and applications to chinese ssec index abstract this paper presents the binomial markov switching multifractal bmsm model of asset returns with skewed t innovations bmsm skewed t for short which considers the fat tails skewness and multifractality in asset returns simultaneously the parameters of bmsm skewed t model can be estimated by maximum likelihood ml methods and volatility forecasting can be accomplished via bayesian updating in order to evaluate the performance of bmsm skewed t model bmsm model with normal innovations bmsm n bmsm model with student t innovations bmsm t and garch 1 1 models garch n garch t and garch skewed t are chosen for comparison through empirical studies on shanghai stock exchange composite index ssec we find that for sample estimation bmsm models outperform the garch 1 1 models through bic and aic rules and bmsm skewed t performs the best among all the models due to its fat tails skewness and multifractality in addition bmsm skewed t model dominates other models at most forecasting horizons for out of sample volatility forecasts in terms of mse mae and spa test

percolation threshold fisher exponent and shortest path exponent for four and five dimensions we develop a method of constructing percolation clusters that allows us to build very large clusters using very little computer memory by limiting the maximum number of sites for which we maintain state information to a number of the order of the number of sites in the largest chemical shell of the cluster being created the memory required to grow a cluster of mass s is of the order of s ensuremath theta bytes where ensuremath theta ranges from 0 4 for two dimensional 2d lattices to 0 5 for six or higher dimensional lattices we use this method to estimate d mathrm min the exponent relating the minimum path mathcal l to the euclidean distance r for 4d and 5d hypercubic lattices analyzing both site and bond percolation we find d mathrm min 1 607 ifmmode pm else textpm fi 0 005 4d and d mathrm min 1 812 ifmmode pm else textpm fi 0 006 5d in order to determine d mathrm min to high precision and without bias it was necessary to first find precise values for the percolation threshold p c p c 0 196889 ifmmode pm else textpm fi 0 000003 4d and p c 0 14081 ifmmode pm else textpm fi 0 00001 5d for site and p c 0 160130 ifmmode pm else textpm fi 0 000003 4d and p c 0 118174 ifmmode pm else textpm fi 0 000004 5d for bond percolation we also calculate the fisher exponent ensuremath tau determined in the course of calculating the values of p c ensuremath tau 2 313 ifmmode pm else textpm fi 0 003 4d and ensuremath tau 2 412 ifmmode pm else textpm fi 0 004 5d

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

percolation in networks with long range connections abstract two dimensional lattices of points are connected with long range links whose lengths are distributed according to p r r by changing the decay exponent one can go from d dimensional short range networks to dimensional networks topologically similar to random graphs percolation on these networks is numerically studied for systems of up to 10 7 sites the shortest path fractal and chemical dimensions are determined at the critical threshold as a function of the decay exponent

series studies of self avoiding walks near the theta point on 2d critical percolation clusters the thermally weighted average end to end distance rn of interacting self avoiding walks saws are obtained enumerating all the finite n stepped saw configurations on the infinite percolation cluster of bond diluted square lattice at the percolation threshold averaging over 250 percolation clusters and enumerating all the possible saws on them for n up to 31 rn is fitted to a scaling form r2n approximately n2 nu theta f nphi tau t theta theta is the temperature interval away from the theta point phi is the crossover exponent and nu theta is the tricritical size exponent the best fit is obtained for theta approximately 0 71 compared to theta 0 approximately 1 54 on a pure square lattice nu approximately 0 74 compared to nu theta phi 4 mod 7 approximately 0 57 in two dimensions and phi approximately 0 20 they also obtain an estimate for of saw size exponent nu c for collapsed phase on the percolation cluster

fractional calculus and continuous time finance ii the waiting time distribution we complement the theory of tick by tick dynamics of financial markets based on a continuous time random walk ctrw model recently proposed by scalas et al 4 and we point out its consistency with the behaviour observed in the waiting time distribution for bund future prices traded at liffe london