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

percolation of interacting classical dimers on the square lattice abstract we study the percolation properties of the interacting classical dimer model on the square lattice by means of monte carlo simulations and finite size scaling analysis we define ising clusters based on the dimer configuration the percolation point of the clusters coincides with the critical point of the kosterlitz thouless transition of the dimer model which is t c 0 654 2 furthermore we find that the largest cluster at the kosterlitz thouless point is a fractal with fractal dimension d c 1 874 2 which coincides with the critical exponent describing the critical behavior of the dimer dimer correlation function which is theoretically predicted to be 15 8

hybrid cluster identification i present a hybrid method for the labelling of clusters in two dimensional lattices which combines the recursive approach with iterative scanning to reduce the stack size required by the pure recursive technique while keeping its benefits single pass and straightforward cluster characterization and percolation detection parallel to the labelling while the capacity to hold the entire lattice in memory is usually regarded as the major constraint for the applicability of the recursive technique the required stack size is the real limiting factor resorting to recursion only for the transverse direction greatly reduces the recursion depth and therefore the required stack it also enhances the overall performance of the recursive technique as is shown by results on a set of uniform random binary lattices and on a set of samples of the ising model i also show how this technique may replace the recursive technique in wolff s cluster algorithm decreasing the risk of stack overflow and increasing its speed and the hoshen kopelman algorithm in the swendsen wang cluster algorithm allowing effortless characterization during generation of the samples and increasing its speed

percolation thresholds and fractal dimensions for square and cubic lattices with long range correlated defects we study long range power law correlated disorder on square and cubic lattices in particular we present high precision results for the percolation thresholds and the fractal dimension of the largest clusters as a function of the correlation strength the correlations are generated using a discrete version of the fourier filtering method we consider two different metrics to set the length scales over which the correlations decay showing that the percolation thresholds are highly sensitive to such system details by contrast we verify that the fractal dimension d f is a universal quantity and unaffected by the choice of metric we also show that for weak correlations its value coincides with that for the uncorrelated system in two dimensions we observe a clear increase of the fractal dimension with increasing correlation strength approaching d f 2 the onset of this change does not seem to be determined by the extended harris criterion

global dynamics of a stochastic neuronal oscillator nonlinear oscillators have been used to model neurons that fire periodically in the absence of input these oscillators which are called neuronal oscillators share some common response structures with other biological oscillations such as cardiac cells in this study we analyze the dependence of the global dynamics of an impulse driven stochastic neuronal oscillator on the relaxation rate to the limit cycle the strength of the intrinsic noise and the impulsive input parameters to do this we use a markov operator that both reflects the density evolution of the oscillator and is an extension of the phase transition curve which describes the phase shift due to a single isolated impulse previously we derived the markov operator for the finite relaxation rate that describes the dynamics of the entire phase plane here we construct a markov operator for the infinite relaxation rate that describes the stochastic dynamics restricted to the limit cycle in both cases the response of the stochastic neuronal oscillator to time varying impulses is described by a product of markov operators furthermore we calculate the number of spikes between two consecutive impulses to relate the dynamics of the oscillator to the number of spikes per unit time and the interspike interval density specifically we analyze the dynamics of the number of spikes per unit time based on the properties of the markov operators each markov operator can be decomposed into stationary and transient components based on the properties of the eigenvalues and eigenfunctions this allows us to evaluate the difference in the number of spikes per unit time between the stationary and transient responses of the oscillator which we show to be based on the dependence of the oscillator on past activity our analysis shows how the duration of the past neuronal activity depends on the relaxation rate the noise strength and the impulsive input parameters

exact enumeration of self avoiding walks on critical percolation clusters in 2 7 dimensions we study self avoiding walks on critical percolation clusters by means of a recently developed exact enumeration method which can handle walks of several thousand steps we had previously presented results for the two and three dimensional cases here we take a wider perspective and vary the system s dimensions up to d 7 beyond the supposed upper critical dimension of these results may serve to check analytical predictions and help understand how the medium s fractal structure impacts on the walks scaling behavior

convergence of threshold estimates for two dimensional percolation using a recently introduced algorithm for simulating percolation in microcanonical fixed occupancy samples we study the convergence with increasing system size of a number of estimates for the percolation threshold for an open system with a square boundary specifically for site percolation on a square lattice we show that the convergence of the average probability estimate is described by a nontrivial correction to scaling exponent as predicted previously and measure the value of this exponent to be 0 90 ifmmode pm else textpm fi 0 02 for the median and cell to cell estimates of the percolation threshold we verify that convergence does not depend on this exponent having instead a slightly faster convergence with a trivial analytic leading exponent

critical exponents for the homology of fortuin kasteleyn clusters on a torus a fortuin kasteleyn cluster on a torus is said to be of type a b a b in z if it is possible to draw a curve belonging to the cluster that winds a times around the first cycle of the torus as it winds b times around the second even though the q potts models make sense only for q integers they can be included into a family of models parametrized by beta square root of q for which the fortuin kasteleyn clusters can be defined for any real beta 0 2 for this family we study the probability pi a b of a given type of clusters as a function of the torus modular parameter tau tau r itau i we compute the asymptotic behavior of some of these probabilities as the torus becomes infinitely thin for example the behavior of pi 1 0 is studied for tau i infinity exponents describing these behaviors are defined and related to weights h r s of the extended kac table for r and s integers but also half integers numerical simulations are also presented possible relationship with recent works and conformal loop ensembles is discussed

percolation on two and three dimensional lattices in this work we apply a highly efficient monte carlo algorithm recently proposed by newman and ziff to treat percolation problems the site and bond percolations are studied on a number of lattices in two and three dimensions quite good results for the wrapping probabilities correlation length critical exponent and critical concentration are obtained for the square simple cubic hexagonal close packed and hexagonal lattices by using relatively small systems we also confirm the universal aspect of the wrapping probabilities regarding site and bond dilution

algorithm to determine the percolation largest component in interconnected networks interconnected networks have been shown to be much more vulnerable to random and targeted failures than isolated ones raising several interesting questions regarding the identification and mitigation of their risk the paradigm to address these questions is the percolation model where the resilience of the system is quantified by the dependence of the size of the largest cluster on the number of failures numerically the major challenge is the identification of this cluster and the calculation of its size here we propose an efficient algorithm to tackle this problem we show that the algorithm scales as o nlogn where n is the number of nodes in the network a significant improvement compared to o n 2 for a greedy algorithm which permits studying much larger networks our new strategy can be applied to any network topology and distribution of interdependencies as well as any sequence of failures

fast monte carlo algorithm for site or bond percolation we describe in detail an efficient algorithm for studying site or bond percolation on any lattice the algorithm can measure an observable quantity in a percolation system for all values of the site or bond occupation probability from zero to one in an amount of time that scales linearly with the size of the system we demonstrate our algorithm by using it to investigate a number of issues in percolation theory including the position of the percolation transition for site percolation on the square lattice the stretched exponential behavior of spanning probabilities away from the critical point and the size of the giant component for site percolation on random graphs

universality of the crossing probability for the potts model for q 1 2 3 4 the universality of the crossing probability pi hs of a system to percolate only in the horizontal direction was investigated numerically by a cluster monte carlo algorithm for the q state potts model for q 2 3 4 and for percolation q 1 we check the percolation through fortuin kasteleyn clusters near the critical point on the square lattice by using representation of the potts model as the correlated site bond percolation model it was shown that probability of a system to percolate only in the horizontal direction pi hs has the universal form pi hs a q q z for q 1 2 3 4 as a function of the scaling variable z b q l 1 nu q p p c q l zeta q here p 1 exp beta is the probability of a bond to be closed a q is the nonuniversal crossing amplitude b q is the nonuniversal metric factor nu q is the correlation length index and zeta q is the additional scaling index the universal function q x approximately equal exp z the nonuniversal scaling factors were found numerically

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

exact solution of site and bond percolation on small world networks we study percolation on small world networks which has been proposed as a simple model of the propagation of disease the occupation probabilities of sites and bonds correspond to the susceptibility of individuals to the disease and the transmissibility of the disease respectively we give an exact solution of the model for both site and bond percolation including the position of the percolation transition at which epidemic behavior sets in the values of the critical exponents governing this transition the mean and variance of the distribution of cluster sizes disease outbreaks below the transition and the size of the giant component epidemic above the transition

the unreasonable effectiveness of tree based theory for networks with clustering we demonstrate that a tree based theory for various dynamical processes operating on static undirected networks yields extremely accurate results for several networks with high levels of clustering we find that such a theory works well as long as the mean intervertex distance l is sufficiently small that is as long as it is close to the value of l in a random network with negligible clustering and the same degree degree correlations we support this hypothesis numerically using both real world networks from various domains and several classes of synthetic clustered networks we present analytical calculations that further support our claim that tree based theories can be accurate for clustered networks provided that the networks are sufficiently small worlds

invaded cluster algorithm for potts models the invaded cluster algorithm a method for simulating phase transitions is described in detail theoretical albeit nonrigorous justification of the method is presented and the algorithm is applied to potts models in two and three dimensions the algorithm is shown to be useful for both first order and continuous transitions and evidently provides an efficient way to distinguish between these possibilities the dynamic properties of the invaded cluster algorithm are studied numerical evidence suggests that the algorithm has no critical slowing for ising models s1063 651x 96 09208 2

critical percolation on the torus we compute the various crossing probabilities defined by r langlands p pouliot and y saint aubin for the critical percolation on the torus

universality class of the two dimensional site diluted ising model in this work we evaluate the probability distribution function of the order parameter for the two dimensional site diluted ising model extensive monte carlo simulations have been performed for different spin concentrations p 0 70 ensuremath le p ensuremath le 1 on square lattices with linear sizes l 20 ensuremath le l ensuremath le 100 the results indicate that the universality class of the diluted ising model seems to be independent of the amount of dilution logarithmic corrections of the finite size critical temperature behavior of the model can also be inferred even for such small lattices

percolation threshold parameters of fluids extensive monte carlo simulations on three qualitatively different model supercritical fluids square well fluid lennard jonesium and primitive water have been performed to examine percolation threshold parameters for continuum correlated models and their relation to general results valid for random lattice models random site percolation simple cubic lattice has therefore been considered as well two different bond criteria the configurational and self bound ones defining a cluster have been used in addition to the percolation threshold occupation probability p c and the percolation threshold fluid density ensuremath rho c the correlation length exponent ensuremath nu and the wrapping probability at the percolation threshold r w c have also been evaluated it is found that parameters ensuremath nu and r w c exhibit not only strong temperature dependence but also unlike the case of lattice systems dependence on the nature of the system considered and the employed definition of the cluster

relaxation properties in a diffusive model of extended objects on a triangular lattice abstract in a preceding paper scepanovic et al j r scepanovic i loncarevic lj budinski petkovic z m jaksic s b vrhovac phys rev e 84 2011 031109 http dx doi org 10 1103 physreve 84 031109 studied the diffusive motion of k mers on the planar triangular lattice among other features of this system we observed that the suppression of rotational motion results in a subdiffusive dynamics on intermediate length and time scales we also confirmed that systems of this kind generally exhibit heterogeneous dynamics here we extend this analysis to objects of various shapes that can be made by self avoiding random walks on a triangular lattice we start by studying the percolation properties of random sequential adsorption of extended objects on a triangular lattice we find that for various objects of the same length the threshold p of more compact shapes exceeds the p of elongated ones at the lower densities of p the long time decay of the self intermediate scattering function sisf is characterized by the kohlrausch williams watts law it is found that near the percolation threshold p the decay of sisf to zero occurs via the power law for sufficiently low wave vectors our results establish that power law divergence of the relaxation time as a function of density occurs at a shape dependent critical density c above the percolation threshold p in the case of k mers the critical density c cannot be distinguished from the closest packing limit c p l 1 for other objects the critical density c is usually below the jamming limit j a m

period doubling in a periodically forced belousov zhabotinsky reaction using an open flow reactor periodically perturbed with light we observe subharmonic frequency locking of the oscillatory belousov zhabotinsky chemical reaction at one sixth the forcing frequency 6 1 over a region of the parameter space of forcing intensity and forcing frequency where the farey sequence dictates we should observe one third the forcing frequency 3 1 in this parameter region the spatial pattern also changes from slowly moving traveling waves to standing waves with a smaller wavelength numerical simulations of the fitzhugh nagumo equations show qualitative agreement with the experimental observations and indicate that the oscillations in the experiment are a result of period doubling

percolation as a dynamical phenomenon abstract we consider a percolation process where the probability p of having one site or bond occupied increases linearly with time we study the total number of clusters as a function of time or p the statistical distribution of jumps in the size of the major cluster as well as the frequency of these jumps we find that both distributions are power laws with different exponents below and above percolation threshold and we discuss these results

binary tree summation monte carlo simulation for potts models abstract in this talk we briefly comment on sweeny and gliozzi methods cluster monte carlo method and recent transition matrix monte carlo for potts models we mostly concentrate on a new algorithm known as binary tree summation some of the most interesting features of this method will be highlighted such as simulating fractional number of potts states as well as offering the partition function and thermodynamic quantities as functions of temperature in a single run

percolation thresholds on planar euclidean relative neighborhood graphs in the present article statistical properties regarding the topology and standard percolation on relative neighborhood graphs rngs for planar sets of points considering the euclidean metric are put under scrutiny rngs belong to the family of proximity graphs i e their edgeset encodes proximity information regarding the close neighbors for the terminal nodes of a given edge therefore they are e g discussed in the context of the construction of backbones for wireless ad hoc networks that guarantee connectedness of all underlying nodes here by means of numerical simulations we determine the asymptotic degree and diameter of rngs and we estimate their bond and site percolation thresholds which were previously conjectured to be nontrivial we compare the results to regular 2d graphs for which the degree is close to that of the rng finally we deduce the common percolation critical exponents from the rng data to verify that the associated universality class is that of standard 2d percolation

critical behavior of the susceptible infected recovered model on a square lattice by means of numerical simulations and epidemic analysis the transition point of the stochastic asynchronous susceptible infected recovered model on a square lattice is found to be c0 0 176 500 510 where c is the probability a chosen infected site spontaneously recovers rather than tries to infect one neighbor this point corresponds to an infection recovery rate of c 1 c0 c0 4 665 713 and a net transmissibility of 1 c0 1 3 c0 0 538 4102 which falls between the rigorous bounds of the site and bond thresholds the critical behavior of the model is consistent with the two dimensional percolation universality class but local growth probabilities differ from those of dynamic percolation cluster growth as is demonstrated explicitly

enhancing robustness and immunization in geographical networks we find that different geographical structures of networks lead to varied percolation thresholds although these networks may have similar abstract topological structures thus strategies for enhancing robustness and immunization of a geographical network are proposed using the generating function formalism we obtain an explicit form of the percolation threshold q c for networks containing arbitrary order cycles for three cycles the dependence of q c on the clustering coefficients is ascertained the analysis substantiates the validity of the strategies with analytical evidence

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

superlinearly scalable noise robustness of redundant coupled dynamical systems we illustrate through theory and numerical simulations that redundant coupled dynamical systems can be extremely robust against local noise in comparison to uncoupled dynamical systems evolving in the same noisy environment previous studies have shown that the noise robustness of redundant coupled dynamical systems is linearly scalable and deviations due to noise can be minimized by increasing the number of coupled units here we demonstrate that the noise robustness can actually be scaled superlinearly if some conditions are met and very high noise robustness can be realized with very few coupled units we discuss these conditions and show that this superlinear scalability depends on the nonlinearity of the individual dynamical units the phenomenon is demonstrated in discrete as well as continuous dynamical systems this superlinear scalability not only provides us an opportunity to exploit the nonlinearity of physical systems without being bogged down by noise but may also help us in understanding the functional role of coupled redundancy found in many biological systems moreover engineers can exploit superlinear noise suppression by starting a coupled system near not necessarily at the appropriate initial condition

force balance percolation we study models of correlated percolation where there are constraints on the occupation of sites that mimic force balance i e for a site to be stable requires occupied neighboring sites in all four compass directions in two dimensions we prove rigorously that p c 1 for the two dimensional models studied numerical data indicate that the force balance percolation transition is discontinuous with a growing crossover length with perhaps the same form as the jamming percolation models suggesting that all models belong to the same universality class with the same underlying mechanism driving the transition in both cases we find a lower bound for the correlation length in the connected phase and that the correlation function does not appear to be a power law at the transition finally we study the dynamics of the culling procedure invoked to obtain the force balance configurations and find a dynamical exponent similar to that found in sandpile models

percolation thresholds for discrete continuous models with nonuniform probabilities of bond formation this work has been partially supported by the iuventus plus programme founded by the polish ministry of science and higher education ip2014 024373 m l acknowledges spanish mineco project foqus fis2013 46768 erc adg osyris eu ip siqs eu strepequam and eu fetproact quic j w has been partially funded by nsf grant dms 131271

percolation on a multifractal scale free planar stochastic lattice and its universality class we investigate site percolation on a weighted planar stochastic lattice wpsl which is a multifractal and whose dual is a scale free network percolation is typically characterized by a threshold value p c at which a transition occurs and by a set of critical exponents which describe the critical behavior of the percolation probability p p mean cluster size s p and the correlation length besides the exponent characterizes the cluster size distribution function n s p c and the fractal dimension d f characterizes the spanning cluster we numerically obtain the value of p c and of all the exponents these results suggest that the percolation on wpsl belong to a separate universality class than on all other planar lattices

crossing on hyperbolic lattices we divide the circular boundary of a hyperbolic lattice into four equal intervals and study the probability of a percolation crossing between an opposite pair as a function of the bond occupation probability p we consider the 7 3 heptagonal enhanced or extended binary tree ebt the ebt dual and the 5 5 pentagonal lattices we find that the crossing probability increases gradually from 0 to 1 as p increases from the lower p l to the upper p u critical values we find bounds and estimates for the values of p l and p u for these lattices and identify the self duality point p corresponding to where the crossing probability equals 1 2 comparison is made with recent numerical and theoretical results

winding clusters in percolation on the torus and the mobius strip using a simulation technique introduced recently we study winding clusters in percolation on the torus and the mobius strip for different aspect ratios the asynchronous parallelization of the simulation makes very large system and sample sizes possible our high accuracy results are fully consistent with predictions from conformal field theory the numerical results for the mobius strip and the number distribution of winding clusters on the torus await theoretical explanation to our knowledge this study is the first of its kind

universality of the excess number of clusters and the crossing probability function in three dimensional percolation extensive monte carlo simulations were performed to evaluate the excess number of clusters and the crossing probability function for three dimensional percolation on the simple cubic s c face centred cubic f c c and body centred cubic b c c lattices systems l l l0 with l0 l were studied for both bond s c f c c b c c and site f c c percolation the excess number of clusters q b per unit length was confirmed to be a universal quantity with a value q b 0 412 likewise the critical crossing probability in the l0 direction with periodic boundary conditions in thell plane was found to follow a universal exponential decay as a function of r d l 0 l for large r simulations were also carried out to find new precise values of the critical thresholds for site percolation on the f c c and b c c lattices yielding pc f c c d 0 199 2365 0 000 0010 pc b c c d 0 245 9615 0 000 0010 we also report the value pc s c d 0 311 6080 0 000 0004 for site percolation

newman ziff algorithm for the bootstrap percolation application to the archimedean lattices abstract we propose very efficient algorithms for the bootstrap percolation and the diffusion percolation models by extending the newman ziff algorithm of the classical percolation m e j newman and r m ziff 2000 27 using these algorithms and the finite size scaling we calculated with high precision the percolation threshold and critical exponents in the eleven two dimensional archimedean lattices we present the condition for the continuous percolation transition in the bootstrap percolation and the diffusion percolation and show that they have the same critical exponents as the classical percolation within error bars in two dimensions we conclude that the bootstrap percolation and the diffusion percolation almost certainly belong to the same universality class as the classical percolation

stochastic phase transition operator in this study a markov operator is introduced that represents the density evolution of an impulse driven stochastic biological oscillator the operator s stochastic kernel is constructed using the asymptotic expansion of stochastic processes instead of solving the fokker planck equation the markov operator is shown to successfully approximate the density evolution of the biological oscillator considered the response of the oscillator to both periodic and time varying impulses can be analyzed using the operator s transient and stationary properties furthermore an unreported stochastic dynamic bifurcation for the biological oscillator is obtained by using the eigenvalues of the product of the markov operators

scaling and universality in the spanning probability for percolation we discuss the spanning percolation probability function for three different spanning rules in general dimensions with both free and periodic boundary conditions our discussion relies on the renormalization group theory which indicates that apart from a few scale factors the scaling functions for the spanning probability are determined by the fixed point and therefore are universal for every system with the same dimensionality spanning rule aspect ratio and boundary conditions for square and rectangular systems we combine this theory with simple relations between the spanning rules and with duality arguments and find strong relations among different derivatives of the spanning function with respect to the scaling variables thus yielding several universal amplitude ratios and allowing a systematic study of the corrections to scaling both singular and analytic in the system size the theoretical predictions are numerically confirmed with excellent accuracy textcopyright 1996 the american physical society

effect of noise on the critical golden mean quasiperiodic dynamics in the circle map abstract scaling regularities are examined associated with effect of additive noise on a critical circle map at the golden mean rotation number we present an improved numerical estimate for the scaling constant of hamm and graham phys rev a46 1992 6323 responsible for the effect of noise 2 3061852653 decrease of the noise amplitude by this number ensures possibility to distinguish one more level of fractal like structure associated with increase of characteristic time scale by the golden mean factor 5 1 2 numeric results demonstrating evidence of the expected scaling are presented e g portraits of the noisy attractors devil s staircase plots lyapunov charts on the parameter plane in different scales

site bond percolation solution to preventing the propagation of p h y t o p h t h o r a zoospores on plantations we propose a strategy based on the site bond percolation to minimize the propagation of phytophthora zoospores on plantations consisting in introducing physical barriers between neighboring plants two clustering processes are distinguished i one of cells with the presence of the pathogen detected on soil analysis and ii that of diseased plants revealed from a visual inspection of the plantation the former is well described by the standard site bond percolation in the latter the percolation threshold is fitted by a tsallis distribution when no barriers are introduced we provide for both cases the formulas for the minimal barrier density to prevent the emergence of the spanning cluster though this work is focused on a specific pathogen the model presented here can also be applied to prevent the spreading of other pathogens that disseminate by other means from one plant to the neighboring ones finally the application of this strategy to three types of commercially important mexican chili plants is also shown

efficient simulation of the random cluster model the simulation of spin models close to critical points of continuous phase transitions is heavily impeded by the occurrence of critical slowing down a number of cluster algorithms usually based on the fortuin kasteleyn representation of the potts model and suitable generalizations for continuous spin models have been used to increase simulation efficiency the first algorithm making use of this representation suggested by sweeny in 1983 has not found widespread adoption due to problems in its implementation however it has been recently shown that it is indeed more efficient in reducing critical slowing down than the more well known algorithm due to swendsen and wang here we present an efficient implementation of sweeny s approach for the random cluster model using recent algorithmic advances in dynamic connectivity algorithms

epidemic outbreaks on random voronoi delaunay triangulations abstract we study epidemic outbreaks on random delaunay triangulations by applying the asynchronous sir susceptible infected removed dynamics coupled to two dimensional voronoi delaunay triangulations in order to investigate the critical behavior of the model we obtain the cluster size distribution by using newman ziff algorithm allowing to simulate random inhomogeneous lattices and measure any desired observable related to percolation we numerically calculate the order parameter defined as the wrapping cluster density the mean cluster size and binder cumulant ratio defined for percolation in order to estimate the epidemic threshold our findings suggest that the system falls into two dimensional dynamic percolation universality class and the quenched random disorder is irrelevant in agreement with results for classical percolation

percolation on hyperbolic lattices the percolation transitions on hyperbolic lattices are investigated numerically using finite size scaling methods the existence of two distinct percolation thresholds is verified at the lower threshold an unbounded cluster appears and reaches from the middle to the boundary this transition is of the same type and has the same finite size scaling properties as the corresponding transition for the cayley tree at the upper threshold on the other hand a single unbounded cluster forms which overwhelms all the others and occupies a finite fraction of the volume as well as of the boundary connections the finite size scaling properties for this upper threshold are different from those of the cayley tree and two of the critical exponents are obtained the results suggest that the percolation transition for the hyperbolic lattices forms a universality class of its own

rigorous confidence intervals for critical probabilities we use the method of balister bollobas and walters random struct algorithms 26 392 2005 to give rigorous 99 9999 confidence intervals for the critical probabilities for site and bond percolation on the 11 archimedean lattices in our computer calculations the emphasis is on simplicity and ease of verification rather than obtaining the best possible results nevertheless we obtain intervals of width at most 0 0005 in all cases

numerical evaluation of the upper critical dimension of percolation in scale free networks we propose numerical methods to evaluate the upper critical dimension d c of random percolation clusters in erd ifmmode mbox h o else h o fi s r enyi networks and in scale free networks with degree distribution mathcal p k ensuremath sim k ensuremath ensuremath lambda where k is the degree of a node and ensuremath lambda is the broadness of the degree distribution our results support the theoretical prediction d c 2 ensuremath lambda ensuremath 1 ensuremath lambda ensuremath 3 for scale free networks with 3l ensuremath lambda l4 and d c 6 for erd ifmmode mbox h o else h o fi s r enyi networks and scale free networks with ensuremath lambda g4 when the removal of nodes is not random but targeted on removing the highest degree nodes we obtain d c 6 for all ensuremath lambda g2 our method also yields a better numerical evaluation of the critical percolation threshold p c for scale free networks our results suggest that the finite size effects increases when ensuremath lambda approaches 3 from above

scaling of cluster heterogeneity in percolation transitions we investigate a critical scaling law for the cluster heterogeneity h in site and bond percolations in d dimensional lattices with d 2 6 the cluster heterogeneity is defined as the number of distinct cluster sizes as an occupation probability p increases the cluster size distribution evolves from a monodisperse distribution to a polydisperse one in the subcritical phase and back to a monodisperse one in the supercritical phase we show analytically that h diverges algebraically approaching the percolation critical point p c as h p p c 1 with the critical exponent associated with the characteristic cluster size interestingly its finite size scaling behavior is governed by a new exponent h 1 d f d where d f is the fractal dimension of the critical percolating cluster and is the correlation length exponent the corresponding scaling variable defines a singular path to the critical point all results are confirmed by numerical simulations

bond percolation for homogeneous two dimensional lattices abstract bond percolation is studied for the three homogeneous two dimensional lattices square lattice sl triangular lattice tl and honeycomb lattice hl an expanding cell technique is used to obtain percolation thresholds and other relevant information for different cell sizes we extend the analysis as to include slightly asymmetric cells in addition to the usual symmetric cells to get more points in the scaling analysis exact percolation functions are obtained for each size then the percolation threshold is obtained by means of two complementary methods one based on the well known renormalization techniques and the other one introduced here which is based upon determining the inflection point of the percolation curves a comparison of the results obtained by these two methods is performed the study includes iterations to extrapolate numerical results towards the thermodynamic limit critical exponents and are obtained values are compared with numerical results and expected theoretical estimations present results show agreement and even improvement in the case of with respect to some numeric values available in the literature comparison tables are provided

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

inverse cascade in a percolation model hierarchical description of time dependent scaling the dynamics of a two dimensional site percolation model on a square lattice is studied using the hierarchical approach introduced by gabrielov et al phys rev e 60 5293 1999 the key elements of the approach are the tree representation of clusters and their coalescence and the horton strahler scheme for cluster ranking accordingly the evolution of the percolation model is considered as a hierarchical inverse cascade of cluster aggregation a three exponent time dependent scaling for the cluster rank distribution is derived using the tokunaga branching constraint and classical results on percolation in terms of cluster masses deviations from the pure scaling are described an empirical constraint on the dynamics of a rank population is reported based on numerical simulations

percolation of the site random cluster model by monte carlo method we propose a site random cluster model by introducing an additional cluster weight in the partition function of the traditional site percolation to simulate the model on a square lattice we combine the color assignation and the swendsen wang methods to design a highly efficient cluster algorithm with a small critical slowing down phenomenon to verify whether or not it is consistent with the bond random cluster model we measure several quantities such as the wrapping probability re the percolating cluster density p and the magnetic susceptibility per site p as well as two exponents such as the thermal exponent yt and the fractal dimension yh of the percolating cluster we find that for different exponents of cluster weight q 1 5 2 2 5 3 3 5 and 4 the numerical estimation of the exponents yt and yh are consistent with the theoretical values the universalities of the site random cluster model and the bond random cluster model are completely identical for larger values of q we find obvious signatures of the first order percolation transition by the histograms and the hysteresis loops of percolating cluster density and the energy per site our results are helpful for the understanding of the percolation of traditional statistical models

random graphs with arbitrary degree distributions and their applications recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the poisson degree distributions that have been widely studied in the past in this paper we develop in detail the theory of random graphs with arbitrary degree distributions in addition to simple undirected unipartite graphs we examine the properties of directed and bipartite graphs among other results we derive exact expressions for the position of the phase transition at which a giant component first forms the mean component size the size of the giant component if there is one the mean number of vertices a certain distance away from a randomly chosen vertex and the average vertex vertex distance within a graph we apply our theory to some real world graphs including the worldwide web and collaboration graphs of scientists and fortune 1000 company directors we demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world while in others there is a measurable discrepancy between theory and reality perhaps indicating the presence of additional social structure in the network that is not captured by the random graph

scaling and percolation in the small world network model in this paper we study the small world network model of watts and strogatz which mimics some aspects of the structure of networks of social interactions we argue that there is one nontrivial length scale in the model analogous to the correlation length in other systems which is well defined in the limit of infinite system size and which diverges continuously as the randomness in the network tends to zero giving a normal critical point in this limit this length scale governs the crossover from large to small world behavior in the model as well as the number of vertices in a neighborhood of given radius on the network we derive the value of the single critical exponent controlling behavior in the critical region and the finite size scaling form for the average vertex vertex distance on the network and using series expansion and pad e approximants find an approximate analytic form for the scaling function we calculate the effective dimension of small world graphs and show that this dimension varies as a function of the length scale on which it is measured in a manner reminiscent of multifractals we also study the problem of site percolation on small world networks as a simple model of disease propagation and derive an approximate expression for the percolation probability at which a giant component of connected vertices first forms in epidemiological terms the point at which an epidemic occurs the typical cluster radius satisfies the expected finite size scaling form with a cluster size exponent close to that for a random graph all our analytic results are confirmed by extensive numerical simulations of the model

scaling corrections site percolation and ising model in three dimensions using finite size scaling techniques we obtain accurate results for critical quantities of the ising model and the site percolation in three dimensions we pay special attention to parametrizing the corrections to scaling which is necessary to bring the systematic errors below the statistical ones

periodic forcing of a limit cycle oscillator fixed points arnold tongues and the global organization of bifurcations the effects of periodic pulsatile stimuli on a nonlinear limit cycle oscillation are analyzed for various relaxation rates to the limit cycle oscillation in the infinite relaxation limit the effects of periodic stimuli are analyzed by consideration of the bifurcations of circle maps in the case of a finite relaxation rate it is necessary to analyze two dimensional maps of the disk however the simple structure of the limit cycle allows us to carry out detailed theoretical and numerical analyses of the dynamics using the brouwer fixed point theorem we show that for any finite nonzero frequency and amplitude of the stimulus there will be a period 1 fixed point associated with a period 1 phase locking we use analytical and numerical methods to determine the stability boundary of the period 1 fixed point as a function of the stimulation frequency amplitude and relaxation rate to the limit cycle these results combined with numerical studies give insight into the changes in the global organization of the phase locking zones as a function of the relaxation rate to the limit cycle

the probability distribution of the percolation threshold in a large system we show that the distribution of the percolation threshold in a large finite system does not converge to a gaussian when the size of the system goes to infinity provided that the two widely accepted definitions of correlation length are equivalent the shape of the distribution is thus directly related to the presence or absence of logarithmic corrections in the power law for the correlation length the result is obtained by estimating the rate of decay of tail of the limiting distribution in terms of the correlation length exponent v all results are rigorously proven in the 2d case generalizations for three dimensions are also discussed

estimation of bond percolation thresholds on the archimedean lattices we give accurate estimates for the bond percolation critical probabilities on seven archimedean lattices for which the critical probabilities are unknown using an algorithm of newman and ziff

explosive site percolation and finite size hysteresis we report the critical point for site percolation for the explosive type for two dimensional square lattices using monte carlo simulations and compare it to the classical well known percolation we use similar algorithms as have been recently reported for bond percolation and networks we calculate the explosive site percolation threshold as p c 0 695 and we find evidence that explosive site percolation surprisingly may belong to a different universality class than bond percolation on lattices providing that the transitions a are continuous and b obey the conventional finite size scaling forms finally we study and compare the direct and reverse processes showing that while the reverse process is different from the direct process for finite size systems the two cases become equivalent in the thermodynamic limit of large l

simple cubic random site percolation thresholds for neighborhoods containing fourth nearest neighbors in this paper random site percolation thresholds for a simple cubic sc lattice with site neighborhoods containing next next next nearest neighbors 4nn are evaluated with monte carlo simulations a recently proposed algorithm with low sampling for percolation thresholds estimation bastas et al arxiv 1411 5834 is implemented for the studies of the top bottom wrapping probability the obtained percolation thresholds are p c 4nn 0 31160 12 p c 4nn nn 0 15040 12 p c 4nn 2nn 0 15950 12 p c 4nn 3nn 0 20490 12 p c 4nn 2nn nn 0 11440 12 p c 4nn 3nn nn 0 11920 12 p c 4nn 3nn 2nn 0 11330 12 and p c 4nn 3nn 2nn nn 0 10000 12 where 3nn 2nn and nn stand for next next nearest neighbors next nearest neighbors and nearest neighbors respectively as an sc lattice with 4nn neighbors may be mapped onto two independent interpenetrated sc lattices but with a lattice constant that is twice as large the percolation threshold p c 4nn is exactly equal to p c nn the simplified method of bastas et al allows for uncertainty of the percolation threshold value p c to be reached similar to that obtained with the classical method but ten times faster

percolation between vacancies in the two dimensional blume capel model using suitable monte carlo methods and finite size scaling we investigate the blume capel model on the square lattice we construct percolation clusters by placing nearest neighbor bonds between vacancies with a variable bond probability pb at the tricritical point we locate the percolation threshold of these vacancy clusters at pbc 0 706 336 at this point we determine the fractal dimension of the vacancy clusters as xf 0 13085 21 160 and the exponent governing the renormalization flow in the pb direction as yp 0 4262 17 40 for bond probability pbpbc the vacancy clusters maintain strong critical correlations the fractal dimension is xf 0 07502 3 40 and the leading correction exponent is yp 0 452 19 40 the above values fit well in the kac table for the tricritical ising model these vacancy clusters have much analogy with those consisting of ising spins of the same sign although the associated quantities and magnetization m are energylike and magnetic quantities respectively however along the critical line of the blume capel model the vacancies are more or less uniformly distributed over the whole lattice in this case no critical percolation correlations are observed in the vacancy clusters at least in the physical region pb 1

are the tails of percolation thresholds gaussians the probability distribution of percolation thresholds in finite lattices was first believed to follow a normal gaussian behaviour with increasing computer power and more efficient simulational techniques this belief turned to a stretched exponential behaviour instead here based on a further improvement of the monte carlo data we show evidence on square lattices that this question is numerically not yet answered at all