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

stochastic mechanics of reciprocal diffusions the dynamics and kinematics of reciprocal diffusions were examined in a previous paper j math phys 34 1846 1993 where it was shown that reciprocal diffusions admit a chain of conservation laws which close after the first two laws for two disjoint subclasses of reciprocal diffusions the markov and quantum diffusions for the case of quantum diffusions the conservation laws are equivalent to schrodinger s equation the markov diffusions were employed by schrodinger sitzungsber preuss akad wiss phys math kl 144 1931 ann inst h poincare 2 269 1932 nelson dynamical theories of brownian motion princeton university princeton nj 1967 quantum fluctuations princeton university princeton nj 1985 and other researchers to develop stochastic formulations of quantum mechanics called stochastic mechanics we propose here an alternative version of stochastic mechanics based on quantum diffusions a procedure is presented for constructing the quantum diffusion associated to a given wave function it is shown that quantum diffusions satisfy the uncertainty principle and have a locality property whereby given two dynamically uncoupled but statistically correlated particles the marginal statistics of each particle depend only on the local fields to which the particle is subjected however like wigner s joint probability distribution for the position and momentum of a particle the finite joint probability densities of quantum diffusions may take negative values

random walks on hierarchical lattices at the percolation threshold random walks on hierarchical lattices are considered at the percolation threshold real space renormalisation methods are employed to obtain the exact fractal dimension of the incipient infinite cluster iic df and the fractal dimension of the effective one dimensional resistance length of the iic dr for some hierarchical lattices the recent conjecture of aharony and stauffer does not hold for diamond and wheatstone bridge hierarchical lattices

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

to what class of fractals does the alexander orbach conjecture apply discussion de la decouverte numerique recente d alexander et orbach selon laquelle pour un agregat infini naissant en percolation le rapport d f d w dimension fractale de l agregat dimension fractale d une marche aleatoire dans l agregat est approximativement independant de d pour d 1

subrepresentations of kronecker representations translated into the language of representations of quivers a challenge in matrix pencil theory is to find sufficient and necessary conditions for a kronecker representation to be a subfactor of another kronecker representation in terms of their kronecker invariants the problem is reduced to a numerical criterion for a kronecker representation to be a subrepresentation of another kronecker representation in terms of their kronecker invariants the key to the problem is the calculation of ranks of matrices over polynomial rings for this a generalization and specialization approach is introduced this approach is applied to provide a numerical criterion for a preprojective resp regular preinjective kronecker representation to be a subrepresentation of another preprojective resp regular preinjective kronecker representation in terms of their kronecker invariants

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

correspondence a note on efficient smoothing for boundary value models abstract the smoothing problem for two point boundary value models is investigated it is shown that by triangularizing the relevant hamiltonian system a new smoothing algorithm can be derived which is twice as fast as that obtained by diagonalizing the hamiltonian

the spectral dimension of random trees we present a simple yet rigorous approach to the determination of the spectral dimension of random trees based on the study of the massless limit of the gaussian model on such trees as a by product we obtain evidence in favour of a new scaling hypothesis for the gaussian model on generic bounded graphs and in favour of a previously conjectured exact relation between spectral and connectivity dimensions on more general tree like structures

on scaling relations in growth models for percolating clusters and diffusion fronts employing analogies between the growth of the incipient percolating cluster and the growth of a diffusion front the authors develop a scaling theory for this latter model their scaling assumptions support the sapoval rosso gouyet conjecture dh 1 1 nu which relates in two dimensions the fractal dimension dh of the hull of the incipient percolating cluster to the correlation length exponent nu in percolation and yield relations between the static and dynamic exponents of the growth of a diffusion front

linear estimation of boundary value stochastic processes part i the role and construction of complementary models this paper presents a substantial extension of the method of complementary models for minimum variance linear estimation introduced by weinert and desai in their important paper 1 specifically the method of complementary models is extended to solve estimation problems for both discrete and continuous parameter linear boundary value stochastic processes in one and higher dimensions a major contribution of this paper is an application of green s identity in deriving a differential operator representation of the estimator to clarify the development and to illustrate the range of applications of our approach two brief examples are provided one is a 1 d discrete two point boundary value process and the other is a 2 d process governed by poisson s equation on the unit disk

rate processes on fractals theory simulations and experiments heterogeneous kinetics are shown to differ drastically from homogeneous kinetics for the elementary reaction a a products we show that the diffusion limited reaction rate is proportional tot h a 2 or to a x whereh 1 d s 2 x 1 2 d s h 2 h 1 andd s is the effective spectral dimension we note that ford d s 1 h 1 2 andx 3 for percolating clustersd s 4 3 h 1 3 andx 5 2 while for dust ds h 1 2 and x 3 scaling arguments supercomputer simulations and experiments give a consistent picture the interplay of energetic and geometric heterogeneity results in fractal like kinetics and is relevant to excitation fusion experiments in porous membranes films and polymeric glasses however in isotopic mixed crystals the geometric fractal nature percolation clusters dominates

resistance and related scaling properties of self avoiding random walks three scaling exponents for random and self avoiding chains are studied by direct renormalisation these describe the end to end resistance of a conducting chain the arc length of the shortest path traceable between the chain ends and the arc length of chain in no loop the scaling behaviours of these quantities are identical to first order in epsilon 4 d d is the space dimension for a long self avoiding chain the authors predict that each one one is proportional to the chain length

some comments on conditionally markov and reciprocal gaussian processes corresp mehr and mcfadden s conditionally markov processes and jamison s reciprocal processes are two independent extensions of slepian s work on a particular stationary gaussian process the relationship between them and their relationship with gauss markov processes are discussed

fractal geometries in decay models results of computer studies of the geometries produced by two distinct decay models are present while one of the models diffusion limited decay results in compact clusters the other random walk decay produces ramified clusters with the hausdorf dimension df equal to 1 75 or 0 03 and 2 34 or 0 03 in two and three dimensions respectively the spectral dimension ds is found to be 1 68 or 0 13 and 1 31 or 0 04 for d 2 and 3

linear systems some algebraic aspects abstract in the past twenty years mathematically minded engineers working in control theory have discovered a number of results of a purely algebraic nature the present article aims at giving a survey of this area for readers interested in linear algebra but without previous knowledge of this topic

gaussian conditionally markov sequences dynamic models and representations of reciprocal and other classes conditionally markov cm sequences are powerful mathematical tools for modeling problems one class of cm sequences is the reciprocal sequence in application we need not only cm dynamic models but also know how to design model parameters models of two important classes of nonsingular gaussian ng cm sequences called cm l and cm f models and a model of the ng reciprocal sequence called reciprocal cm l model were presented in our previous works and their applications were discussed in this paper these models are studied in more detail in particular their parameter design it is shown that every reciprocal cm l model can be induced by a markov model then parameters of each reciprocal cm l model can be obtained from those of the markov model also it is shown that an ng cm l cm f sequence can be represented by a sum of an ng markov sequence and an uncorrelated ng vector this necessary and sufficient representation provides a basis for designing parameters of a cm l cm f model from the cm viewpoint a representation is also obtained for ng reciprocal sequences this representation is simple and reveals an important property of reciprocal sequences as a result the significance of studying reciprocal sequences from the cm viewpoint is demonstrated a full spectrum of dynamic models from a cm l model to a reciprocal cm l model is also presented some examples are presented for illustration

gaussian conditionally markov sequences singular nonsingular most existing results about modeling and characterizing gaussian markov reciprocal and conditionally markov cm processes assume nonsingularity of the processes this assumption makes the analysis easier but restricts application of these processes this paper studies models and characterizes the general singular nonsingular gaussian cm including reciprocal and markov sequence for example to our knowledge there is no dynamic model for the general singular nonsingular gaussian reciprocal sequence in the literature we obtain two such models from the cm viewpoint as a result the significance of studying reciprocal sequences from the cm viewpoint is demonstrated the results of this paper unify singular and nonsingular gaussian cm including reciprocal and markov sequences and provide tools for their application an application of cm sequences in trajectory modeling with a destination is discussed and illustrative examples are presented

feedback invariants of a pair of matrices with prescribed columns abstract this paper studies the possible feedback equivalence classes of a pair of matrices with some prescribed columns and related problems

combinatorics of polynomial chains abstract in this paper we consider two polynomial chains and we study the existence of an intermediate polynomial chain satisfying interlacing inequality relations together with some convexity like properties that arise from applications in completion problems the presented result is a vast generalisation of existing results dealing with intermediate polynomial chains the proof of the main result is completely novel to the field different from all the existing methods used in similar problems the obtained necessary and sufficient conditions are simple explicit and constructive

structure of clusters generated by random walks the authors study the cluster structure resulting from a nearest neighbour random walk embedded in a d dimensional space each bond visited by the random walks is regarded as belonging to the cluster the diffusion exponent and the fracton dimensional of the fractal cluster in d 3 is found to be dw 3 5 or 0 1 and d 0 57 or 0 02 using a method of exact enumeration of random walks on these fractals

discrete time complementary models and smoothing abstract the fact that the smoothing error is a wide sense markov process is somehow surprising since smoothed estimates depend upon both past and future data in this paper we first give a simple and general proof of this fact then we use the so called complementary models introduced by weinert and desai to derive forwards and backwards markovian models for the smoothing error in state space models by exploring the structure of the complementary models we show that under certain restrictions only two simple structured models exist one that runs forwards in time and the other that runs backwards in time the forwards complementary model leads to the forward rauch tung striebel rts smoothing formula and to a backwards markovian model for the error whereas the backwards model leads to the backward rts formula and to a forwards error model the two models for the smoothing error can be derived one from the other by a forward backward transformation that preserves the sample paths finally by using

stationarity and reciprocity in stochastic multipoint boundary value systems investigates certain statistical properties of the state vector in a stochastic multipoint boundary value system the authors give necessary and sufficient conditions for the state vector to be stationary and prove that the state vector is locally reciprocal

anomalous diffusion on regular and random models for diffusion limited aggregation families of regular fractal models for witten sander dla clusters in d dimensions are proposed resembling the computer generated aggregates the authors evaluate random walk dimensionalities exactly using resistance scaling ideas the applicability of the conflicting conjectures of alexander orbach 1982 ao dw 3df 2 and aharony stauffer 1984 as dw df 1 to dla is then examined one finds that the dendritic nature of the fractals presented ensures that as is satisfied exactly in all dimensions an rsrg fugacity transformation method is applied to the simplest of the regular fractals in d 2 the result for dw is compared with the exact resistance scaling result to obtain an estimate of the error introduced by the rsrg the rg method is then generalised to treat the random dla problem the resulting estimate for dw is in fortuitously good agreement with as the corresponding monte carlo data and the results above for the regular fractal models

structural and dynamical properties of random walk clusters the authors study the structural and dynamical properties of the clusters generated by a nearest neighbour random walk embedded in a d dimensional space they have focused on the non trivial case in which the cluster is generated in d 3 the structure of this cluster is characterised by loops for all length scales on the one hand and by the fact that deadends are negligible upon scaling on the other hand the cluster is very dilute and is characterised by fractal dimension df 2 and chemical dimension d1 1 29 or 0 04 from these results it follows that v identical to dldf approximately 2 3 which is consistent with the formula v 2 d 2 or d or 4 obtained using a flory type argument the dynamical diffusion exponents dw and dlw were calculated using the exact enumeration method and found to be dw 3 45 or 0 10 and dlw 2 28 or 0 05 the results suggest that the effect of loops is small but not negligible they also calculated the fracton dimensionality of the cluster and obtained ds 1 14 or 0 02 a scaling function is presented for the end to end mean square displacement of a random walk performed on a random walk cluster this scaling function is supported by their numerical results

representations for multivariate reciprocal gaussian processes multivariate reciprocal gaussian processes are represented as a sum of two independent processes a piecewise markov process which is also represented in terms of a wiener type process and a time dependent linear transformation of a normally distributed random vector this result is then applied to the first passage time problem

interlacing inequalities for regular pencils abstract we solve the problem of finding a necessary and sufficient condition for the existence of a regular pencil when its invariant homogeneous factors and a subpencil are prescribed the solution obtained is an extension of the well known sa thompson interlacing inequalities for invariant factors of characteristic matrices

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

modeling and estimation of discrete time gaussian reciprocal processes discrete time gaussian reciprocal processes are characterized in terms of a second order two point boundary value nearest neighbor model driven by a locally correlated noise whose correlation is specified by the model dynamics this second order model is the analog for reciprocal processes of the standard first order state space models for markov processes the model is used to obtain a solution to the smoothing problem for reciprocal processes the resulting smoother obeys second order equations whose structure is similar to that of the kalman filter for gauss markov processes it is shown that the smoothing error is itself a reciprocal process

convection enhanced diffusion for random flows we analyze the effective diffusivity of a passive scalar in a two dimensional steady incompressible random flow that has mean zero and a stationary stream function we show that in the limit of small diffusivity or large peclet number with convection dominating there is substantial enhancement of the effective diffusivity our analysis is based on some new variational principles for convection diffusion problems and on some facts from continuum percolation theory some of which are widely believed to be correct but have not been proved yet we show in detail how the variational principles convert information about the geometry of the level lines of the random stream function into properties of the effective diffusivity and substantiate the result of isichenko and kalda that the effective diffusivity behaves like 3 13 when the molecular diffusivity is small assuming some percolation theoretic facts we also analyze the effective diffusivity for a special class of convective flows random cellular flows where the facts from percolation theory are well established and their use in the variational principles is more direct than for general random flows

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

modelling and estimation for finite state reciprocal processes a reciprocal equation is a kind of descriptor linear discrete index stochastic system which is well known be satisfied pathwise by all gaussian reciprocal processes from a system theoretic point of view it is a kind of noncausal linear system in the sense that the solution of it cannot be determined by only an initial condition indeed requiring the terminal state as well besides all the input function between initial and terminal states also nice properties are known of a reciprocal equation such as the equivalence of it with a couple of ordinary causal dynamic systems running in opposite directions for these reasons here we assume a reciprocal equation as the target of stochastic realization for the class of finite state reciprocal processes also named reciprocal chains the central result of the present paper is showing that any canonical reciprocal chain i e valued in the canonical base of realrn n being the cardinality of the set of chain s states satisfies pathwise a reciprocal equation in a n2 dimensional canonical variable or in other word a quadratic reciprocal equation named augmented state reciprocal model asrm also for a partially observed reciprocal chain a linear optimal smoother is derived all the results here presented are based upon the idea that a reciprocal chain is a combination of markov bridges to this purpose other forms besides the asrm are presented in order to make clear the meaning of this combination as well as to prove that the linear smoother can be actually implemented as n smoothers all operating independently on each markov bridge component

matrix pencils completion problems ii in this paper we give new simple and explicit necessary and sufficient conditions for the existence of a matrix pencil with prescribed kronecker invariants and a regular subpencil moreover we explicitly solve the problem of describing the possible kronecker invariants of a pencil with a prescribed subpencil in the case of the existence of an intermediate regular pencil all the results are obtained over algebraically closed fields

smoothing and likelihood ratio for gaussian boundary value processes a new derivation which does not need the invertibility assumption of the covariance matrix of the boundary data is given for the smoothing of gaussian two point boundary value processes tpbvp the likelihood ratio for tpbv processes is then derived in terms of the system parameters by using the krein factorization the likelihood ratio involves the smoother of the process an alternate expression for the likelihood ratio based on the filtered estimate of the state is also given

pencils with prescribed constant subpencils abstract the results in this paper are within the scope of matrix pencil completion problems given an arbitrary matrix pencil we obtain necessary and sufficient conditions for the existence of a strictly equivalent pencil with a prescribed constant subpencil for algebraically closed fields

singular systems state feedback problem in this paper the problem of kronecker invariants assignment by state feedback in singular linear systems is studied and resolved this result presents a generalization of the previous results of state feedback action on singular systems

random walk on the high dimensional iic we study the asymptotic behavior of the exit times of random walk from euclidean balls around the origin of the incipient infinite cluster in a manner inspired by kumagai and misumi j theor probab 21 910 935 2008 we do this by getting bounds on the effective resistance between the origin and the boundary of these euclidean balls we show that the geometric properties of long range percolation clusters are significantly different from those of finite range clusters we also study the behavior of random walk on the backbone of the iic and we prove that the alexander orbach conjecture holds for the incipient infinite cluster in high dimensions both for long range percolation and for finite range percolation

a new characterization for multivariate gaussian reciprocal processes established a new necessary and sufficient condition for a multivariate gaussian random process to be reciprocal this result is then used to prove that the state vector of a two point boundary value system is reciprocal the authors do not impose any nonsingularity conditions on the covariance function of the state vector

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

maximum likelihood sequence estimation for hidden reciprocal processes this paper addresses the problem of maximum likelihood sequence estimation mlse based on a hidden reciprocal chain hrc as the underlying target model hrcs are non causal discrete time finite state stochastic processes which can be regarded as the one dimensional version of a markov random field although they are not in general markov processes this paper describes a procedure for evaluating the mlse for hrc and compares the resultant estimator with its markov model equivalent the viterbi algorithm in addition the performance of the newly proposed reciprocal mlse is compared to a hrc based optimal smoother

completion of quasi regular matrix pencils abstract in this paper we give solutions for two important particular cases of the general completion problem involving quasi regular matrix pencil we describe the possible kronecker invariants of a quasi regular pencil whose subpencil is prescribed in the case when the prescribed subpencil is without nontrivial homogeneous invariant factors also we solve the dual problem by describing the possible kronecker invariants of a pencil without nontrivial homogeneous invariant factors when a quasi regular subpencil is prescribed these two results are expected to be the crucial steps toward a solution of the general completion problem the results are given over algebraically closed fields

the fractal dimension of cluster perimeters generated by a kinetic walk the fractal dimension dg of cluster perimeters generated by a recently proposed butterfly growth walk is considered in the long range limit of the walk on a percolation cluster dg appears to be equal to the fractal dimension of the singly connected bonds dg 1 nu the new relation for chemical dimension dl is proposed dl df df dg in the short range limit the butterfly walk on a euclidean lattice appears to be in the same universality class as a random walk the dynamic aspect of the growth walk is discussed and the continuously tunable spectral dimension is obtained both short and long range limits of this diffusion process are different from a random diffusion on percolation

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

unified theorems on completions of matrix pencils abstract we generalize and unify some theorems concerned with the existence of a pencil with a prescribed subpencil and prescribed strict equivalence class

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

the incipient infinite cluster in two dimensional percolation letpp be the probability measure on the configurations of occupied and vacant vertices of a two dimensional graphg under which all vertices are independently occupied respectively vacant with probabilityp respectively 1 p letph be the critical probability for this system andw the occupied cluster of some fixed vertexw0 we show that for many graphsg such as mathbb z 2 or its covering graph which corresponds to bond percolation on mathbb z 2 the following two conditional probability measures converge and have the same limit v say

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

assigning the kronecker invariants of a matrix pencil by row or column completions abstract the challenge consists in describing the relationships between the kronecker invariants of a matrix pencil and one of its subpencils for a given subpencil an algorithm for constructing a matrix pencil with prescribed kronecker invariants should also be proposed

surface structure and anisotropy of eden clusters the simple eden model is simulated with clusters which are orders of magnitude larger than those of some previous work the surface perimeter is slightly anisotropic and feels the underlying structure of the square lattice even for 17 million cluster sites the width of the surface layer increases with increasing cluster mass with an effective exponent presumably larger than 1 4 for averages over the whole circumference and perhaps equal to 1 4 for flat surfaces width averages over a small angle are compared with the recent prediction of plischke and racz 1985 even 17 million sites are not enough to see directly the asymptotic behaviour

minimax and interlacing thoerems for matrices abstract an exposition is given of some recent and important work by e marques de sa and r c thompson which characterized the relationship between the similarity invariants of a square matrix over a field and those of a principal submatrix of that matrix this work is then related to results on eigenvalues of complex hermitian matrices the courant fischer minimax theorem and the cauchy interlacing theorem on singular values of rectangular complex matrices due to thompson and on invariant factors of rectangular matrices over a principal ideal domain due to sa and thompson a partial unification of these theories is presented and several open questions stated

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

high dimensional incipient infinite clusters revisited the incipient infinite cluster iic measure is the percolation measure at criticality conditioned on the cluster of the origin to be infinite using the lace expansion we construct the iic measure for high dimensional percolation models in three different ways extending previous work by the second named author and jarai we show that each construction yields the same measure indicating that the iic is a robust object furthermore our constructions apply to spread out versions of both finite range and long range percolation models we also get estimates on structural properties of the iic such as the volume of the intersection between the iic and euclidean balls