a note on differentiability of the cluster density for independent percolation in high dimensions the cluster density function of independent percolation in ad dimensional lattice is considered for eachn it is shown that p has finitenth leftderivative at critical probabilityp c ifd is sufficiently large this result agrees with the bethe lattice approximation where thenth one sided derivative of p is bounded atp c for alln

the alexander orbach conjecture holds in high dimensions we examine the incipient infinite cluster iic of critical percolation in regimes where mean field behavior has been established namely when the dimension d is large enough or when d 6 and the lattice is sufficiently spread out we find that random walk on the iic exhibits anomalous diffusion with the spectral dimension d s frac 4 3 that is p t x x t 2 3 o 1 this establishes a conjecture of alexander and orbach j phys lett paris 43 625 631 1982 en route we calculate the one arm exponent with respect to the intrinsic distance

the survival probability and r point functions in high dimensions in this paper we investigate the survival probability n in high dimensional statistical physical models where n denotes the probability that the model survives up to time n we prove that if the r point functions scale to those of the canonical measure of super brownian motion and if certain self repellence and total population tail bound conditions are satised then n n 2 av where a is the asymptotic expected number of particles alive at timen andv is the vertex factor of the model our results apply to spread out lattice trees above 8 dimensions spread out oriented percolation above 4 1 dimensions and the spread out contact process above 4 1 dimensions in the case of oriented percolation this reproves a result by the rst author den hollander and slade which was proved using heavy lace expansion arguments at the cost of losing explicit error estimates we further derive several consequences of our result involving the scaling limit of the number of particles alive at time proportional to n our proofs are based on simple weak convergence arguments

the scaling limit of the incipient infinite cluster in high dimensional percolation ii integrated super brownian excursion for independent nearest neighbor bond percolation on zd with d 6 we prove that the incipient infinite cluster s two point function and three point function converge to those of integrated super brownian excursion ise in the scaling limit the proof is based on an extension of the new expansion for percolation derived in a previous paper and involves treating the magnetic field as a complex variable a special case of our result for the two point function implies that the probability that the cluster of the origin consists of n sites at the critical point is given by a multiple of n 3 2 plus an error term of order n 3 2 e with e 0 this is a strong version of the statement that the critical exponent is given by 2

mean field behavior for the survival probability and the percolation point to surface connectivity we consider the critical survival probability up to time t for oriented percolation and the contact process and the point to surface of the ball of radius t connectivity for critical percolation let t denote both quantities we prove in a unifled fashion that if t exhibits a power law and both the two point function and its certain restricted version exhibit the same mean fleld behavior then t t i1 for the time oriented models with d 4 and t t i2 for percolation with d 7

tree graph inequalities and critical behavior in percolation models various inequalities are derived and used for the study of the critical behavior in independent percolation models in particular we consider the critical exponent associated with the expected cluster sizex and the structure of then site connection probabilities n x1 xn it is shown that quite generally 1 the upper critical dimension above which attains the bethe lattice value 1 is characterized both in terms of the geometry of incipient clusters and a diagramatic convergence condition for homogeneousd dimensional lattices with x y o x y d 2 atp p c our criterion shows that 1 if 6 d 3 the connectivity functions n are generally bounded by tree diagrams which involve the two point function we conjecture that above the critical dimension the asymptotic behavior of n in the critical regime is actually given by such tree diagrams modified by a nonsingular vertex factor other results deal with the exponential decay of the cluster size distribution and the function 2 x y

random packings of spheres built with sequential models several models have been introduced to construct random packings of identical spheres the packings are built on a horizontal basal layer using iterative sequential algorithms in which a new added sphere is always in contact with three other spheres in the packing in the bennet model the position with the lowest vertical coordinate is selected in the anti bennet model only positions that are stable under gravity are considered and that with the highest vertical coordinate is selected in the eden model the new sphere is selected at random among all the possibilities and in the stable eden model the random choice is limited to positions that are stable under gravity in all cases the density packing fraction is determined within an uncertainty of or 10 4 the histogram for the number of contacts between spheres the bond angle distribution and the distance distribution were also determined the results are compared with corresponding ballistic model results

incipient infinite percolation clusters in 2d we study several kinds of large critical percolation clusters in two dimensions we show that from the microscopic lattice scale perspective these clusters can be described by kesten s incipient infinite cluster iic as was conjectured by aizenman more specifically we establish this for incipient spanning clusters large clusters in a finite box and the inhomogeneous model of chayes chayes and durrett our results prove the equivalence of several natural definitions of the iic we also show that for any k 1 the difference in size between the kth and k 1 st largest critical clusters in a finite box goes to infinity in probability as the size of the box goes to infinity in addition the distribution of the chayes chayes durrett cluster is shown to be singular with respect to the iic

infinite canonical super brownian motion and scaling limits we construct a measure valued markov process which we call infinite canonical super brownian motion and which corresponds to the canonical measure of super brownian motion conditioned on non extinction infinite canonical super brownian motion is a natural candidate for the scaling limit of various random branching objects on mathbb z d when these objects are critical mean field and infinite we prove that icsbm is the scaling limit of the spread out oriented percolation incipient infinite cluster above 4 dimensions and of incipient infinite branching random walk in any dimension we conjecture that it also arises as the scaling limit in various other models above the upper critical dimension such as the incipient infinite lattice tree above 8 dimensions the incipient infinite cluster for unoriented percolation above 6 dimensions uniform spanning trees above 4 dimensions and invasion percolation above 6 dimensions

rigorous analysis of discontinuous phase transitions via mean field bounds we consider a variety of nearest neighbor spin models defined on the d dimensional hypercubic lattice z d our essential assumption is that these models satisfy the condition of reflection positivity we prove that whenever the associated mean field theory predicts a discontinuous transition the actual model also undergoes a discontinuous transition which occurs near the mean field transition temperature provided the dimension is sufficiently large or the first order transition in the mean field model is sufficiently strong as an application of our general theory we show that for d sufficiently large the 3 state potts ferromagnet on z d undergoes a first order phase transition as the temperature varies similar results are established for all q state potts models with q 3 the r component cubic models with r 4 and the o n nematic liquid crystal models with n 3

random graph asymptotics on high dimensional tori we investigate the scaling of the largest critical percolation cluster on a large d dimensional torus for nearest neighbor percolation in sufficiently high dimensions or when d 6 for sufficiently spread out percolation we use a relatively simple coupling argument to show that this largest critical cluster is with high probability bounded above by a large constant times v 2 3 and below by a small constant times v 2 3 log v 4 3 where v is the volume of the torus we also give a simple criterion in terms of the subcritical percolation two point function on mathbb z d under which the lower bound can be improved to small constant times v 2 3 i e we prove random graph asymptotics for the largest critical cluster on the high dimensional torus this establishes a conjecture by 1 apart from logarithmic corrections we discuss implications of these results on the dependence on boundary conditions for high dimensional percolation

mean field critical behaviour for correlation length for percolation in high dimensions extending the method of 27 we prove that the corrlation length of independent bond percolation models exhibits mean field type critical behaviour i e p p c p 1 2 asp p c in two situations i for nearest neighbour independent bond percolation models on ad dimensional hypercubic lattice d withd sufficiently large and ii for a class of spread out independent bond percolation models which are believed to belong to the same universality class as the nearest neighbour model in more than six dimensions the proof is based on and extends a method developed in 27 where it was used to prove the triangle condition and hence mean field behaviour of the critical exponents and 2 for the above two cases

concerning maximal packing arrangements of binary disk mixtures the determination of the maximal packing arrangements of two dimensional binary hard disks of radii rs and rl with rs 6 rl for su0ciently small rs amounts to 1nding the optimal arrangement of the small disks within a tricusp the nonconvexcavity between three close packed large disks we present a particle growth monte carlo algorithm for the generation of geometric packings of equi sized hard disks within such a tricusp the 1rst 19 members of an in1nite sequence of maximal density structures thus produced are reported in addition the monte carlo algorithm is applied to the geometric packing of disks within a 8at sided equilateral triangle and compared to published results for that packing problem we perform an analysis of geometric properties of the packings e g packing fractions and symmetries of structures con1ned to both containers interestingly we 1nd a non monotonic increase in the packing fraction with increasing number of disks packed within both the 8at sided triangle and tricusp it is important to note that for disk packings within a 8at sided equilateral triangle this non monotonic behavior of the packing fraction had not been reported in previously published works for the 8at sided equilateral triangle local maxima occur at the triangular integers ns 1 3 6 10 15 as well as ns 1 2 where ns is the number of disks in each packing however local maxima for packings within the tricusp exist at ns 1 3 6 10 18 finally we analyze the asymptotic approach to the upper bound on the packing fraction of the in1nite sequence of maximal structures of disks con1ned to the tricusp c

space filling tiling by random packing of discs abstract we study here a space filling tiling generated by random non overlapping packing of discs to place the n th disc we randomly choose a point in the empty pore space of the n 1 discs and place a disc with the centre at this point and touching the nearest disc we measure the fractal dimension of the pore space left in the asymptotic limit for these space filling patterns

multiple phase transitions in long range first passage percolation on square lattices we consider a model of long range first passage percolation on the d dimensional square lattice in which any two distinct vertices are connected by an edge having exponentially distributed passage time with mean where 0 is a fixed parameter and is the l1 norm on we analyze the asymptotic growth rate of the set t which consists of all such that the first passage time between the origin 0 and x is at most t as t we show that depending on the values of there are four growth regimes i instantaneous growth for 2d 1 like the nearest neighbor first passage percolation model corresponding to 2015 wiley periodicals inc

the uniform random tree in a brownian excursion to any brownian excursione with duration e and anyt 1 t p 0 e we associate a branching tree withp branches denoted byt p e t 1 t p which is closely related to the structure of the minima ofe our main theorem states that ife is chosen according to the ito measure and t 1 t p according to lebesgue measure on 0 e p the treet p e t 1 t p is distributed according to the uniform measure on the set of trees withp branches the proof of this result yields additional information about the subexcursions ofe corresponding to the different branches of the tree thus generalizing a well known representation theorem of bismut if we replace the ito measure by the law of the normalized excursion a simple conditioning argument leads to another remarkable result originally proved by aldous with a very different method

gap exponents for percolation processes with triangle condition a study is made of the gap exponents for percolation processes with the triangle condition in the subcritical region it is show that the gaps are given by t 2 fort 2 3 scaling theory predicts thatp p c 0 s p p c p ande p 1 c 0 c 0 s p p c p 3 wheres p is the typical cluster size it is found that p c p p p c 0 s p 1 p c p 1 2 and p c p 3 e p 1 c 0 c 0 s p 1 p c p 3 4

the incipient infinite cluster for high dimensional unoriented percolation we consider bond percolation on mathbb z d at the critical occupation density p c for d 6 in two different models the first is the nearest neighbor model in dimension d 6 the second model is a spread out model having long range parameterized by l in dimension d 6 in the spread out case we show that the cluster of the origin conditioned to contain the site x weakly converges to an infinite cluster as x when d 6 and l is sufficiently large we also give a general criterion for this convergence to hold which is satisfied in the case d 6 in the nearest neighbor model by work of hara 12 we further give a second construction by taking p 6 the probability for invasion percolation to reach a site x is asymptotic to c x d 4 as x

geometric properties of random disk packings random packings ofn 2000 rigid disks in the plane subject to periodic boundary conditions on a square primitive cell have been generated by a concurrent construction which treats all disks on an equal footing as opposed to previously investigated sequential constructions the particles start with random positions and velocities and as they move about they grow uniformly in size from points to jammed disks the collection of packings displays several striking geometric features these include for largen typically polycrystalline textures with irregular grain boundaries and linear shear fractures the packings occasionally contain monovacancies and trapped but unjammed rattler disks the latter appear to be confined to the grain boundaries the linear shear fractures preserve bond orientational order but disrupt translational order within the crystalline grains a new efficient event driven simulation algorithm is employed to generate the histories of colliding and jamming disks on a computer which can process one million floating point instructions per second the algorithm processes more than one million pairwise collisions per hour

on the scaling of the chemical distance in long range percolation models we consider the unoriented long range percolation on z d in dimensions d 1 where distinct sites x y z d get connected with probability p xy 0 1 assuming p xy x y s o 1 as x y where s 0 and is a norm distance on z d and supposing that the resulting random graph contains an infinite connected component c we let d x y be the graph distance between x and y measured on c our main result is that for s d 2d d x y log x y o 1 x y c x y where 1 is the binary logarithm of 2d s and o 1 is a quantity tending to zero in probability as x y besides its interest for general percolation theory this result sheds some light on a question that has recently surfaced in the context of small world phenomena as part of the proof we also establish tight bounds on the probability that the largest connected component in a finite box contains a positive fraction of all sites in the box

a meshless local kriging method for large deformation analyses in this paper a well developed meshless numerical technique local kriging lokriging method is used to develop the lokriging formulation for the large deformation problems which are geometrically nonlinear this is the first work for the geometrically nonlinear analyses by this meshless local weak form method in the lokriging method the kriging interpolation is employed to construct the meshless shape functions the spline function with high continuity is used as the weight function the discrete equations of the large deformation analysis for two dimensional solids are obtained using the local weak forms and based on the total lagrangian tl approach the presently developed lokriging method is a truly meshless method as it does not require the mesh either for the construction of the shape functions or for the integration of the local weak form several numerical examples of the two dimensional geometrically nonlinear analysis are presented to illustrate the performance of the present lokriging method the present nonlinear lokriging formulation is also used for the large deformation analysis of the microelectromechanical systems mems device the numerical results are compared with the experimental results very good results are obtained all these applications have demonstrated that the present lokriging method is very effective for the large deformation analyses because it avoids all mesh distortion issues

an efficient dense and stable particular elements generation method based on geometry summary in this paper a new discrete elements generation method based on geometry is proposed to fill geometric domains with particles disks or spheres by generating particles each one with a random radius or with a radius calculated from the iteration to ensure no overlaps exist between particles and identifying unstable particles and changing them to stable ones a dense and stable packing can be created a partitioning particle radius interval method and a particle stability inspection and improvement method are introduced to guarantee the algorithm s success and the stability of the particles some packings were created to evaluate the performances of the new method the results showed that the algorithm was very efficient and was able to create isotropic packings of low porosities and large coordinate numbers the partitioning particle radius interval method improved the generation efficiency significantly and increased the packing densities through the comparisons with several existing methods proposed recently the method proposed in this work is found to be more efficient and can fill geometric domains with the lowest porosities in addition the stability of the particles is guaranteed and no complex triangular or tetrahedral mesh is required in particle generation thereby making the new method simpler copyright 2016 john wiley sons ltd

inhomogeneous percolation problems and incipient infinite clusters the authors consider inhomogeneous percolation models with density pc f x and examine the forms of f x which produce incipient structures taking f x approximately mod x mod lambda and assuming the existence of a correlation length exponent v for the homogeneous percolation model they prove that in d 2 the borderline value of lambda is lambda b 1 v if lambda 1 v then with probability one there is no infinite cluster while if lambda 2 the models studied suggest what sort of incipient objects should be examined in random surface models

the scaling limit of lattice trees in high dimensions we prove that above eight dimensions the scaling limit of sufficiently spread out lattice trees is the variant of super brownian motion known as integrated super brownian excursion ise as conjectured by aldous the same is true for nearest neighbour lattice trees in sufficiently high dimensions the proof uses the lace expansion

geometrical modeling of granular structures in two and three dimensions application to nanostructures a granular structure can be modeled by a parallelepiped containing spherical balls in three dimensions or by a rectangle filled with disks in two dimensions these grains spherical balls or disks are separated by interfaces called grain boundaries and their size correspond to a size distribution which is obtained experimentally the geometrical modeling of such a structure consists in determining the repartition of the set of disjoint grains according to these specifications in this paper a new constructive algorithm based on an advancing front approach usually used in the context of mesh generation is proposed this algorithm is nearly linear in complexity robust and fast in both two and three dimensions enhancements in computing time and density are observed and reported via comparisons with existing methods moreover we propose a method to transform spherical balls disks into polyhedral polygonal cells similar to the real grain shapes examples of nanostructure modeling in two and three dimensions are presented copyright 2009 john wiley sons ltd

a power counting theorem for feynman integrals on the lattice a convergence theorem is proved which states sufficient conditions for the existence of the continuum limit for a wide class of feynman integrals on a space time lattice a new kind of a uv divergence degree is introcduced which allows the formulation of the theorem in terms of power counting conditions

tree based models for random distribution of mass a mathematical model for distribution of mass ind dimensional space based upon randomly embedding random trees into space is introduced and studied the model is a variant of thesuper brownian motion process studied by mathematicians we present calculations relating to i the distribution of position of a typical mass element ii moments of the center of mass iii large deviation behavior and iv a recursive self similarity property

packing spherical discrete elements for large scale simulations abstract we introduce a new geometric method to generate sphere packings with restricted overlap values sample generation is an important but time consuming step that precedes a calculation performed with the discrete element method dem at present there does not exist any software dedicated to dem which would be similar to the mesh software that exists for finite element methods fem a practical objective of the method is to build very large sphere packings several hundreds of thousands in a few minutes instead of several days as the current dynamic methods do the developed algorithm uses a new geometric procedure to position very efficiently the polydisperse spheres in a tetrahedral mesh the algorithm implemented into yade open dem open source software consists in filling tetrahedral meshes with spheres in addition to the features of the tetrahedral mesh the input parameters are the minimum and maximum radii or their size ratio and the magnitude of authorized overlaps the filling procedure is stopped when a target solid fraction or number of spheres is reached based on this method an efficient tool can be designed for dems used by researchers and engineers the generated packings can be isotropic and the number of contacts per sphere is very high due to its geometric procedure in this paper different properties of the generated packings are characterized and examples from real industrial problems are presented to show how this method can be used the current c version of this packing algorithm is part of yade open dem 20 available on the web https yade dem org

the scaling limit of the incipient infinite cluster in high dimensional percolation i critical exponents this is the first of two papers on the critical behavior of bond percolation models in high dimensions in this paper we obtain strong joint control of the critical exponents and for the nearest neighbor model in very high dimensions d 6 and for sufficiently spread out models in all dimensions d 6 the exponent describes the low frequency behavior of the fourier transform of the critical two point connectivity function while describes the behavior of the magnetization at the critical point our main result is an asymptotic relation showing that in a joint sense 0 and 2 the proof uses a major extension of our earlier expansion method for percolation this result provides evidence that the scaling limit of the incipient infinite cluster is the random probability measure on d known as integrated super brownian excursion ise in dimensions above 6 in the sequel to this paper we extend our methods to prove that the scaling limits of the incipient infinite cluster s two point and three point functions are those of ise for the nearest neighbor model in dimensions d 6

a convergence theorem for lattice feynman integrals with massless propagators it is shown that for non vanishing lattice spacing conventional infrared power counting conditions are sufficient for convergence of lattice feynman integrals with zero mass propagators if these conditions are supplemented by ultraviolet convergence conditions the continuum limit of such a diagram exists and is universal

the number and size of branched polymers in high dimensions we consider two models of branched polymers lattice trees on thed dimensional hypercubic lattice i the nearest neighbor model in sufficiently high dimensions and ii a spread out or long range model ford 8 in which trees are constructed from bonds of length less than or equal to a large parameterl we prove that for either model the critical exponent for the number of branched polymers exists and equals 5 2 and that the critical exponentv for the radius of gyration exists and equals 1 4 this improves our earlier results for the corresponding generating functions the proof uses the lace expansion together with an analysis involving fractional derivatives which has been applied previously to the self avoiding walk in a similar context

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

computer simulation of random packing of unequal particles a monte carlo simulation model for the random packing of unequal spherical particles is presented in this paper with this model the particle radii obeying a given distribution are generated and randomly placed within a cubic packing domain with high packing density and many overlaps then a relaxation iteration is applied to reduce or eliminate the overlaps while the packing space is gradually expanded the simulation is completed once the mean overlap value falls below a preset value to simulate the random close packing a vibration process is applied after the relaxation iteration for log normal distributed particles the effect of particle size standard deviation and for bidisperse particles the effects of particle size ratio and the volume fraction of large particles on packing density and on coordination number are investigated simulation results show good agreement with that obtained by experiments and by other simulations the randomness homogeneity and isotropy which have not been evaluated before for packing of distributed particles are also examined using statistical measures

mean field critical behaviour for percolation in high dimensions the triangle condition for percolation states that sum limits x y tau 0 x tau 0 y cdot tau y 0 is finite at the critical point where x y is the probability that the sitesx andy are connected we use an expansion related to the lace expansion for a self avoiding walk to prove that the triangle condition is satisfied in two situations i for nearest neighbour independent bond percolation on thed dimensional hypercubic lattice ifd is sufficiently large and ii in more than six dimensions for a class of spread out models of independent bond percolation which are believed to be in the same universality class as the nearest neighbour model the class of models in ii includes the case where the bond occupation probability is constant for bonds of length less than some large number and is zero otherwise in the course of the proof an infrared bound is obtained the triangle condition is known to imply that various critical exponents take their mean field bethe lattice values gamma beta 1 delta delta t 2 t underset raise0 3em hbox smash scriptscriptstyle geqslant 2 and that the percolation density is continuous at the critical point we also prove thatv2 in i and ii wherev2 is the critical exponent for the correlation length

percolation critical exponents under the triangle condition for independent percolation models it is shown that if the diagrammatic triangle condition is satisfied then the critical exponents delta and hat beta exist and take their mean field values generalizing the criterion introduced in 1984 by aizenman and newman for the mean field value of gamma in nonoriented percolation the results apply to a broad class of nonoriented as well as oriented weakly homogeneous models in which the range of the connecting bonds need not be bounded for the nonoriented case the condition reduces to the finiteness at the critical point of nabla equiv sum x y tau 0 x tau x y tau y 0 with tau u nu the probability that the site u is connected to nu which was recently established by hara and slade for models with sufficiently spread out connections in d 6 dimensions our analysis proceeds through the derivation of complementary differential inequalities for the percolation order parameter m beta h whose value at h 0 yields the percolation density with beta parametrizing the bond or site occupation probabilities and with h h geq 0 a ghost field the conclusion is that under the triangle condition in the vicinity of the critical point beta c 0 m beta 0 approx beta beta c hat beta and m beta c h approx h 1 delta with hat beta 1 and delta 2