quenched invariance principle for random walks in balanced random environment we consider random walks in a balanced random environment in mathbb z d d 2 we first prove an invariance principle for d 2 and the transience of the random walks when d 3 recurrence when d 2 in an ergodic environment which is not uniformly elliptic but satisfies certain moment condition then using percolation arguments we show that under mere ellipticity the above results hold for random walks in i i d balanced environments

consensus of a group of mobile agents in three dimensions abstract this paper investigates the consensus problem of a three dimensional 3d version of the widely studied vicsek model where each member moves in 3d euclidean space with a constant speed with headings updated according to the average direction of neighbors in comparison with the original vicsek model each agent s heading is determined by two angle sequences interacting with each other one of which evolves according to a linear non homogeneous equation which makes the theoretical analysis quite complicated by analyzing the underlying structure of the system and relying on estimation of some characteristics concerning the initial states we will establish consensus results for 3d vicsek model under random framework without resorting to any a priori connectivity assumption

a subdivision algorithm for the computation of unstable manifolds and global attractors each invariant set of a given dynamical system is part of the global attractor therefore the global attractor contains all the potentially interesting dynamics and in particular it contains every global unstable manifold for this reason it is of interest to have an algorithm which allows to approximate the global attractor numerically in this article we develop such an algorithm using a subdivision technique we prove convergence of this method in a very general setting and moreover we describe the qualitative convergence behavior in the presence of a hyperbolic structure the algorithm can successfully be applied to dynamical systems of moderate dimension and we illustrate this fact by several numerical examples

quenched invariance principles for random walks on percolation clusters we consider a supercritical bernoulli percolation model in d2 and study the simple symmetric random walk on the infinite percolation cluster the aim of this paper is to prove the almost sure q

simple random walk on long range percolation clusters ii scaling limits supported in part at the technion by a landau fellowship supported in part by an alfred sloan fellowship in mathematics

lower bounds on the proportion of leaders needed for expected consensus of 3 d flocks this paper considers the consensus behavior of a spatially distributed 3 d dynamical network composed of heterogeneous agents leaders and followers in which the leaders have the preferred information about the destination while the followers do not have all followers move in a 3 d euclidean space with a given speed and with their headings updated according to the average velocity of the corresponding neighbors compared with the 2 d model a key point lies in how to analyze the dynamical behavior of a linear nonhomogeneous equation where the nonhomogeneous term strongly nonlinearly depends on the states of all agents using the network structure and the estimation of some characteristics for the initial states we present a proper decaying rate for the nonhomogeneous term and then establish lower bounds on the ratio of the number of leaders to the number of followers that is needed for the expected consensus by considering two cases 1 fixed speed and neighborhood radius and 2 variable speed and neighborhood radius with respect to the population size some simulation examples are given to justify the theoretical results

the inflation of attractors and their discretization the autonomous case the effects of perturbation or discretization on a maximal attractor of autonomous dynamical systems is now well understood with the perturbed or discretized system having a nearby maximal attractor 5 7 10 in contrast for nonautonomous systems the very concept of a nonautonomous attractor itself is still undergoing intensive development and perturbation or discretization results have been obtained this work was supported by the dfg forschungschwerpunkt ergodentheorie analysis und effiziente simulation dynamischer systeme the author was partially supported by the russian foundation for basic researches grant 97 01 00692

random walks on supercritical percolation clusters we obtain gaussian upper and lower bounds on the transition density qt x y of the continuous time simple random walk on a supercritical percolation cluster c1 in the euclidean lattice the bounds analogous to aronsen s bounds for uniformly elliptic divergence form diusions hold with constants ci depending only on p the percolation probability and d the irregular nature of the medium means that the bound for qt x only holds for t sx where the constant sx depends on the percolation congura tion

stability of multiagent systems with time dependent communication links we study a simple but compelling model of network of agents interacting via time dependent communication links the model finds application in a variety of fields including synchronization swarming and distributed decision making in the model each agent updates his current state based upon the current information received from neighboring agents necessary and or sufficient conditions for the convergence of the individual agents states to a common value are presented thereby extending recent results reported in the literature the stability analysis is based upon a blend of graph theoretic and system theoretic tools with the notion of convexity playing a central role the analysis is integrated within a formal framework of set valued lyapunov theory which may be of independent interest among others it is observed that more communication does not necessarily lead to faster convergence and may eventually even lead to a loss of convergence even for the simple models discussed in the present paper

greedy lattice animals geometry and criticality assign to each site of the integer lattice z d a real score sampled according to the same distribution f independently of the choices made at all other sites a lattice animal is a finite connected set of sites with its weight being the sum of the scores at its sites let n n be the maximal weight of those lattice animals of size n that contain the origin denote by n the almost sure finite constant limit of n 1 n n which exists under a mild condition on the positive tail of f we study certain geometrical aspects of the lattice animal with maximal weight among those contained in an n box where n is large both in the supercritical phase where n 0 and in the critical case where n 0

synchronization reachable topology and synchronization of discrete time linear multi agent systems in this note we consider the synchronization problem for discrete time linear multi agent systems with time varying network topology to describe a new joint connectivity condition for synchronization the concept of synchronization reachable topology is introduced under the conditions that agent dynamics is neutrally stable and network topology is synchronization reachable a synchronization and an exponential synchronization result are obtained using paracontraction theory the two results generalize in a natural way the classical results for the consensus of first order discrete time multi agent systems finally the effectiveness of the results is illustrated by a simulation example

the smallest possible interaction radius for synchronization of self propelled particles a central and fundamental issue in the theory of complex systems is to understand how local rules lead to collective behavior of the whole system this paper will investigate a typical collective behavior synchronization of a self propelled particle system modeled by the nearest neighbor rules while connectivity of the dynamic neighbor graphs associated with the underlying systems is crucial for synchronization it is widely known that the verification of such dynamical connectivity is at the core of theoretical analysis ideally conditions used for synchronization should be imposed on the model parameters and the initial states of the particles one crucial model parameter is the interaction radius and we are interested in the following natural and basic question what is the smallest interaction radius for synchronization in this paper we will show that in a certain sense the smallest possible interaction radius approximately equals sqrt log n pi n with n being the population size w

invariance principle for the random conductance model we study a continuous time random walk x in an environment of i i d random conductances mu e in 0 infty in mathbb z d we assume that mathbb p mu e 0 p c so that the bonds with strictly positive conductances percolate but make no other assumptions on the law of the e we prove a quenched invariance principle for x and obtain green s functions bounds and an elliptic harnack inequality

on symmetric random walks with random conductances on z d we study models of continuous time symmetric z d valued random walks in random environments one of our aims is to derive estimates on the decay of transition probabilities in a case where a uniform ellipticity assumption is absent we consider the case of independent conductances with a polynomial tail near 0 and obtain precise asymptotics for the annealed return probability and convergence times for the random walk confined to a finite box

stable attracting sets in dynamical systems and in their one step discretizations we consider a dynamical system described by a system of ordinary differential equations which possesses a compact attracting set of arbitrary shape under the assumption of uniform asymptotic stability of in the sense of lyapunov we show that discretized versions of the dynamical system involving one step numerical methods have nearby attracting sets h which are also uniformly asymptotically stable our proof uses the properties of a lyapunov function which characterizes the stability of

synchronization of multi agent systems without connectivity assumptions multi agent systems arise from diverse fields in natural and artificial systems such as schooling of fish flocking of birds coordination of autonomous agents in multi agent systems a typical and basic situation is the case where each agent has the tendency to behave as other agents do in its neighborhood through computer simulations vicsek et al 1995 showed that such a simple local interaction rule can lead to a certain kind of cooperative phenomenon synchronization of the overall system if the initial states are randomly distributed and the size of the system population is large since this model is of fundamental importance in understanding the multi agent systems it has attracted much research attention in recent years in this paper we will present a comprehensive theoretical analysis for this class of multi agent systems under a random framework with large population but without imposing any connectivity assumptions as did in almost all of the previous investigations to be precise we will show that for any given and fixed model parameters concerning with the interaction radius r and the agents moving speed v the overall system will synchronize as long as the population size n is large enough furthermore to keep the synchronization property as the population size n increases both r and v can actually be allowed to decrease according to certain scaling rates

the smallest possible interaction radius for flock synchronization this paper investigates the synchronization behavior of a class of flocks modeled by the nearest neighbor rules while connectivity of the associated dynamical neighbor graphs is crucial for synchronization it is well known that the verification of such dynamical connectivity is the core of theoretical analysis ideally conditions used for synchronization should be imposed on the model parameters and the initial states of the agents one crucial model parameter is the interaction radius and we are interested in the following natural but complicated question what is the smallest interaction radius for synchronization of flocks in this paper we reveal that in a certain sense the smallest possible interaction radius approximately equals log n n with n being the population size which coincides with the critical radius for connectivity of random geometric graphs given by gupta and kumar critical power for asymptotic connectivity in wireless networks in stochastic analysis control optimization and applications birkhauser boston boston ma 1999 pp 547 566

asymptotic stability equals exponential stability and iss equals finite energy gain if you twist your eyes in this paper we show that uniformly global asymptotic stability for a family of ordinary differential equations is equivalent to uniformly global exponential stability under a suitable nonlinear change of variables the same is shown for input to state stability and input to state exponential stability and for input to state exponential stability and a nonlinear h estimate

existence of the harmonic measure for random walks on graphs and in random environments we give a sufficient condition for the existence of the harmonic measure from infinity of transient random walks on weighted graphs in particular this condition is verified by the random conductance model on d d 3 when the conductances are i i d and the bonds with positive conductance percolate the harmonic measure from infinity also exists for random walks on supercritical clusters of 2 this is proved using results of barlow ann probab 32 3024 3084 2004 and barlow and hambly electron j probab 14 1 1 27 2009

quenched invariance principle for simple random walk on percolation clusters we consider the simple random walk on the unique infinite cluster of super critical bond percolation in d with d 2 we prove that for almost every percolation configuration the path distribution of the walk converges weakly to that of non degenerate isotropic brownian motion our analysis is based on the consideration of a harmonic deformation of the infinite cluster on which the random walk becomes a square integrable martingale the size of the deformation expressed by the so called corrector is estimated by means of ergodicity arguments

quenched invariance principles for walks on clusters of percolation or among random conductances in this work we principally study random walk on the supercritical infinite cluster for bond percolation on d we prove a quenched functional central limit theorem for the walk when d 4 we also prove a similar result for random walk among i i d random conductances along nearest neighbor edges of d when d 1

quenched invariance principle for simple random walk on clusters in correlated percolation models we prove a quenched invariance principle for simple random walk on the unique infinite percolation cluster for a general class of percolation models on mathbb z d d ge 2 with long range correlations introduced in drewitz et al in j math phys 55 8 083307 2014 solving one of the open problems from there this gives new results for random interlacements in dimension d ge 3 at every level as well as for the vacant set of random interlacements and the level sets of the gaussian free field in the regime of the so called local uniqueness which is believed to coincide with the whole supercritical regime an essential ingredient of our proof is a new isoperimetric inequality for correlated percolation models

quenched invariance principle for random walks with time dependent ergodic degenerate weights we study a continuous time random walk xx on zdzd in an environment of dynamic random conductances taking values in 0 0 we assume that the law of the conductances is ergodic with respect to space time shifts we prove a quenched invariance principle for the markov process xx under some moment conditions on the environment the key result on the sublinearity of the corrector is obtained by moser s iteration scheme

a matrix approach to graph maximum stable set and coloring problems with application to multi agent systems using the semi tensor product of matrices this paper investigates the maximum weight stable set and vertex coloring problems of graphs with application to the group consensus of multi agent systems and presents a number of new results and algorithms firstly by defining a characteristic logical vector and using the matrix expression of logical functions an algebraic description is obtained for the internally stable set problem based on which a new algorithm to find all the internally stable sets is established for any graph secondly the maximum weight stable set problem is considered and a necessary and sufficient condition is presented by which an algorithm to find all the maximum weight stable sets is obtained thirdly the vertex coloring problem is studied by using the semi tensor product method and two necessary and sufficient conditions are proposed for the colorability based on which a new algorithm to find all the k coloring schemes and minimum coloring partitions is put forward finally the obtained results are applied to multi agent systems and a new protocol design procedure is presented for the group consensus problem the study of illustrative examples shows that the results algorithms presented in this paper are very effective

isoperimetry and heat kernel decay on percolation clusters we prove that the heat kernel on the infinite bernoulli percolation cluster in zd almost surely decays faster than t d 2 we also derive estimates on the mixing time for the random walk confined to a finite box our approach is based on local isoperimetric inequalities some of the results of this paper were previously announced in the note of mathieu and remy c r acad sci paris ser i math 332 2001 927 931

a necessary and sufficient condition for consensus over random networks we consider the consensus problem for stochastic discrete time linear dynamical systems the underlying graph of such systems at a given time instance is derived from a random graph process independent of other time instances for such a framework we present a necessary and sufficient condition for almost sure asymptotic consensus using simple ergodicity and probabilistic arguments this easily verifiable condition uses the spectrum of the average weight matrix finally we investigate a special case for which the linear dynamical system converges to a fixed vector with probability 1

evolving sets mixing and heat kernel bounds we show that a new probabilistic technique recently introduced by the first author yields the sharpest bounds obtained to date on mixing times of markov chains in terms of isoperimetric properties of the state space also known as conductance bounds or cheeger inequalities we prove that the bounds for mixing time in total variation obtained by lovasz and kannan can be refined to apply to the maximum relative deviation p n x y y 1 of the distribution at time n from the stationary distribution we then extend our results to markov chains on infinite state spaces and to continuous time chains our approach yields a direct link between isoperimetric inequalities and heat kernel bounds previously this link rested on analytic estimates known as nash inequalities

invariance principle for the random conductance model in a degenerate ergodic environment we study a continuous time random walk x on zd in an environment of random conductances taking values in 0 we assume that the law of the conductances is ergodic with respect to space shifts we prove a quenched invariance principle for x under some moment conditions of the environment the key result on the sublinearity of the corrector is obtained by moser s iteration scheme

on the security of public key algorithms based on chebyshev polynomials over the finite field z_n in this paper the period distribution of sequences generated by chebyshev polynomials over the finite field zn is analyzed it is found that the distribution is unsatisfactory if n the modulus is not chosen properly based on this finding we present an attack on the public key algorithm based on chebyshev polynomials over zn then we modify the original algorithm to make it suitable for practical purpose its security under some existing models is also discussed in detail

on the anisotropic walk on the supercritical percolation cluster we investigate in this work the asymptotic behavior of an anisotropic random walk on the supercritical cluster for bond percolation on d d 2 in particular we show that for small anisotropy the walk behaves in a ballistic fashion whereas for strong anisotropy the walk is sub diffusive for arbitrary anisotropy we also prove the directional transience of the walk and construct a renewal structure

cucker smale flocking under rooted leadership with free will agents abstract motivated by the recent works of cucker smale and shen we study the emergent behavior of discrete time cucker smale flocking under rooted leadership with free will agents we prove that if the free will accelerations of the agents relative to the leader are summable then the individual velocities will converge to the velocity of leader under suitable initial conditions in particular we prove that the unconditional convergence is true if the decay exponent of the communication weight is below some critical value which depends on the interaction topology we also present some numerical examples to illustrate the main result

isoperimetry in two dimensional percolation we study isoperimetric sets i e sets with minimal boundary for a prescribed volume on the unique infinite connected component of supercritical bond percolation on the square lattice in the limit of the volume tending to infinity properly scaled isoperimetric sets are shown to converge in the hausdorff metric to the solution of an isoperimetric problem in 2 with respect to a particular norm as part of the proof we also show that the anchored isoperimetric profile as well as the cheeger constant of the giant component in finite boxes scale to deterministic quantities this settles a conjecture of itai benjamini for the square lattice 2015 wiley periodicals inc

convergence rates of perturbed attracting sets with vanishing perturbation we investigate the rate of convergence and the rate of attraction of perturbed attracting sets as the perturbation tends to zero the perturbation model under consideration is a control system which contains the unperturbed system as zero dynamics necessary and sufficient conditions for certain rates of convergence are derived several applications especially to numerical approximations are given and a number of examples illustrate the results

consensus seeking in multiagent systems under dynamically changing interaction topologies this note considers the problem of information consensus among multiple agents in the presence of limited and unreliable information exchange with dynamically changing interaction topologies both discrete and continuous update schemes are proposed for information consensus this note shows that information consensus under dynamically changing interaction topologies can be achieved asymptotically if the union of the directed interaction graphs have a spanning tree frequently enough as the system evolves

quenched invariance principles for random walks with random conductances we prove an almost sure invariance principle for a random walker among i i d conductances in d d 2 we assume conductances are bounded from above but we do not require that they are bounded from below

public key cryptosystems based on cubic finite field extensions the cryptographic properties of third order linear feedback shift register lfsr sequences over gf p are investigated a fast computational algorithm for evaluating the kth term of a characteristic sequence of order 3 is presented based on these properties a new public key distribution scheme and an rsa type encryption algorithm are proposed their security implementation information rate and computational cost for the new schemes are discussed

on chemical distances and shape theorems in percolation models with long range correlations in this paper we provide general conditions on a one parameter family of random infinite subsets of z d to contain a unique infinite connected component for which the chemical distances are comparable to the euclidean distance in addition we show that these conditions also imply a shape theorem for the corresponding infinite connected component by verifying these conditions for specific models we obtain novel results about the structure of the infinite connected component of the vacant set of random interlacements and the level sets of the gaussian free field as a byproduct we obtain alternative proofs to the corresponding results for random interlacements in the work of cern and popov on the internal distance in the interlacement set electron j probab 17 29 1 25 2012 and while our main interest is in percolation models with long range correlations we also recover results in the spirit of the work of antal and pisztora on the chemical distance for supercritical bernoulli percolation ann probab 24 2 1036 1048 1996 for bernoulli percolation finally as a corollary we derive new results about the chemical diameter of the largest connected component in the complement of the trace of the random walk on the torus

simple random walk on long range percolation clusters i heat kernel bounds in this paper we derive upper bounds for the heat kernel of the simple random walk on the infinite cluster of a supercritical long range percolation process for any d 1 and for any exponent s in d d 2 wedge 2d giving the rate of decay of the percolation process we show that the return probability decays like t d s d up to logarithmic corrections where t denotes the time the walk is run our methods also yield generalized bounds on the spectral gap of the dynamics and on the diameter of the largest component in a box the bounds and accompanying understanding of the geometry of the cluster play a crucial role in the companion paper crawford and sly in simple randomwalk on long range percolation clusters ii scaling limit 2010 where we establish the scaling limit of the random walk to be stable levy motion

anchored nash inequalities and heat kernel bounds for static and dynamic degenerate environments abstract we introduce anchored versions of the nash inequality they allow to control the l 2 norm of a function by dirichlet forms that are not uniformly elliptic we then use them to provide heat kernel upper bounds for diffusions in degenerate static and dynamic random environments as an example we apply our results to the case of a random walk with degenerate jump rates that depend on an underlying exclusion process at equilibrium

mixing time of near critical random graphs let mathcal c 1 be the largest component of the erd s renyi random graph mathcal g n p the mixing time of random walk on mathcal c 1 in the strictly supercritical regime p c n with fixed c 1 was shown to have order log2n by fountoulakis and reed and independently by benjamini kozma and wormald in the critical window p 1 e n where e3n is bounded nachmias and peres proved that the mixing time on mathcal c 1 is of order n however it was unclear how to interpolate between these results and estimate the mixing time as the giant component emerges from the critical window indeed even the asymptotics of the diameter of mathcal c 1 in this regime were only recently obtained by riordan and wormald as well as the present authors and kim in this paper we show that for p 1 e n with e3n and o n the mixing time on mathcal c 1 is with high probability of order n log2 in addition we show that this is the order of the largest mixing time over all components both in the slightly supercritical and in the slightly subcritical regime i e p 1 e n with as above

emergent behavior in flocks we provide a model for both continuous and discrete time describing the evolution of a flock our model is parameterized by a constant beta capturing the rate of decay which in our model is polynomial of the influence between birds in the flock as they separate in space our main result shows that when beta 1 2 convergence of the flock to a common velocity is guaranteed while for betages1 2 convergence is guaranteed under some condition on the initial positions and velocities of the birds only

trapping in the random conductance model we consider random walks on d among nearest neighbor random conductances which are i i d positive bounded uniformly from above but whose support extends all the way to zero our focus is on the detailed properties of the paths of the random walk conditioned to return back to the starting point at time 2n we show that in the situations when the heat kernel exhibits subdiffusive decay which is known to occur in dimensions d 4 the walk gets trapped for a time of order n in a small spatial region this shows that the strategy used earlier to infer subdiffusive lower bounds on the heat kernel in specific examples is in fact dominant in addition we settle a conjecture concerning the worst possible subdiffusive decay in four dimensions

mixing time for random walk on supercritical dynamical percolation we consider dynamical percolation on the d dimensional discrete torus znd of side length n where each edge refreshes its status at rate n 1 2 to be open with probability p we study random walk on the torus where the walker moves at rate 1 2d along each open edge in earlier work of two of the authors with a stauffer it was shown that in the subcritical case p pc zd the mixing time is n2 1 here the implied constants depend only on d and p we prove a quenched and hence annealed version of this conjecture up to a poly logarithmic factor under the assumption p 1 2 when p 0 we prove a version of this conjecture for an alternative notion of mixing time involving randomised stopping times the latter implies sharp up to poly logarithmic factors upper bounds on exit times of large balls throughout the supercritical regime our proofs are based on percolation results e g the grimmett marstrand theorem and an analysis of the volume biased evolving set process the key point is that typically the evolving set has a substantial intersection with the giant percolation cluster at many times this allows us to use precise isoperimetric properties of the cluster due to g pete to infer rapid growth of the evolving set which in turn yields the upper bound on the mixing time

adaptive tracking games for coupled stochastic linear multi agent systems stability optimality and robustness distributed adaptive tracking type games are investigated for a class of coupled stochastic linear multi agent systems with uncertainties of unknown structure parameters external stochastic disturbances unmodeled dynamics and unknown agents interactions the control goal is to make the states of all the agents converge to a desired function of the population state average psa due to the fact that only local information is available for each agent the control is distributed for the time invariant parameter case the extended least squares algorithm nash certainty equivalence nce principle and certainty equivalence ce principle are used to estimate the unknown parameters and the psa term and to design adaptive control respectively under some mild conditions it is shown that the closed loop system is almost surely uniformly stable with respect to the population number n the estimate for the psa term is strongly consistent the adaptive control is almost surely an asymptotic nash equilibrium when the dynamics of each agent contains time varying parameters and unmodeled dynamics the projected least mean square lms algorithm nce principle and ce principle are adopted to estimate the unknown time varying parameters and the unknown psa term and to design robust adaptive control respectively in addition to stability of the closed loop system and consistency of the psa estimate the control law is shown to be robust nash equilibrium with respect to the unmodeled dynamics the variation of the unknown parameters and the external disturbances two numerical examples are given to illustrate the methods and results of this paper

synchronization and convergence of linear dynamics in random directed networks recently methods in stochastic control are used to study the synchronization properties of a nonautonomous discrete time linear system x k 1 g k x k where the matrices g k are derived from a random graph process the purpose of this note is to extend this analysis to directed graphs and more general random graph processes rather than using lyapunov type methods we use results from the theory of inhomogeneous markov chains in our analysis these results have been used successfully in deterministic consensus problems and we show that they are useful for these problems as well sufficient conditions are derived that depend on the types of graphs that have nonvanishing probabilities for instance if a scrambling graph occurs with nonzero probability then the system synchronizes

security analysis of the public key algorithm based on chebyshev polynomials over the integer ring zn recently kocarev and tasev 20 proposed to use chebyshev polynomials over real numbers to design a public key algorithm by employing the semigroup property bergamo et al 4 pointed out that the public key algorithm based on chebyshev polynomials working on real numbers is not secure and devised an attack which permits to recover the corresponding plaintext from a given ciphertext later kocarev et al 19 generalized the chebyshev polynomials from real number fields to finite fields and finite rings to make the public key algorithm more secure and practical however we analyzed the period distribution of the sequences generated by the chebyshev polynomials over finite fields 21 when the modulus n is prime we found this algorithm was also not secure and proposed an attack on this algorithm over finite fields we then proposed some schemes to improve the security in this paper we further analyze in detail the period distribution of the sequences generated by chebyshev polynomials over the integer ring z n when n is composite it turns out that the period distribution is poor if n is not chosen properly and there are many small periods which are not secure in the sense of cryptology based on these findings we devise an attack on the public key algorithm based on chebyshev polynomials over the integer ring z n we also propose some suggestions to avoid this attack

random conductance models with stable like jumps heat kernel estimates and harnack inequalities abstract we establish two sided heat kernel estimates for random conductance models with non uniformly elliptic possibly degenerate stable like jumps on graphs these are long range counterparts of the well known two sided gaussian heat kernel estimates by m t barlow for nearest neighbor short range random walks on the supercritical percolation cluster unlike the cases of nearest neighbor conductance models we cannot use parabolic harnack inequalities since even elliptic harnack inequalities do not hold in the present setting as an application we establish the local limit theorem for the models