the brownian web arratia arratia r 1979 ph d thesis university of wisconsin madison and unpublished work and later toth and werner toth b werner w 1998 probab theory relat fields111 375 452 constructed random processes that formally correspond to coalescing one dimensional brownian motions starting from every space time point we extend their work by constructing and characterizing what we call the brownian web as a random variable taking values in an appropriate metric space whose points are compact sets of paths this leads to general convergence criteria and in particular to convergence in distribution of coalescing random walks in the scaling limit to the brownian web

stochastic lattice gas model for a predator prey system we propose a stochastic lattice gas model to describe the dynamics of two animal species population one being a predator and the other a prey this model comprehends the mechanisms of the lotka volterra model our analysis was performed by using a dynamical mean field approximation and computer simulations our results show that the system exhibits an oscillatory behavior of the population densities of prey and predators for the sets of parameters used in our computer simulations these oscillations occur at a local level mean field results predict synchronized collective oscillations

spatial models for species area curves the relationship between species number and area is an old problem in biology we propose here an interacting particle system the multitype voter model with mutation as a mathematical model to study this problem we analyze the species area curves of this model as the mutation rate a tends to zero we obtain two basic types of behavior depending on the size of the spatial region under consideration if the region is a square with area r r 1 then for small the number of species is of order l r log 2 whereas if r 1 the number of species is bounded

quantum operators in classical probability theory iv quasi duality and thinnings of interacting particle systems duality has proved to be a powerful technique in the study of interacting particle systems ips this concept can be enlarged and a quasi duality defined between various pairs of ips previously thought unrelated consequently theorems of a similar style to those involving duality can be deduced the concept of quasi duality follows naturally from our previous studies into the use ofsingle site operators an idea borrowed from quantum physics in paper ii of this series it is shown that a necessary condition for quasi duality is that the eigenvalues of the corresponding two site infinitesimal generators be the same and using this observation a number of quasi dual pairs have been found and studied it is further shown that if two different ips share a common dual then one can be considered as a thinning of the other

critical attractive spin systems we study a class of attractive spin systems we prove that for these processes the system dies out when the parameters are on the critical surface we also prove that the supercritical process survives with positive probability in a sufficiently thick space time slab

residual p properties of mapping class groups and surface groups let m p be the mapping class group of a punctured oriented surface e p where p may be empty and let t p p be the kernel of the action of m p on h 1 p f p we prove that t p p is residually p in particular this shows that m p is virtually residually p for a group g we denote by ip g the kernel of the natural action of out g on h 1 g f p in order to achieve our theorem we prove that under certain conditions g is conjugacy p separable and has property a the group ip g is residually p the fact that free groups and surface groups have property a is due to grossman the fact that free groups are conjugacy p separable is due to lyndon and schupp the fact that surface groups are conjugacy p separable is from a technical point of view the main result of the paper

hitting probability of a distant point for the voter model started with a single 1 the goal of this work is to find the asymptotics of the hitting probability of a distant point for the voter model on the integer lattice started from a single 1 at the origin in dimensions d 2 or 3 we obtain the precise asymptotic behavior of this probability we use the scaling limit of the voter model started from a single 1 at the origin in terms of super brownian motion under its excursion measure this invariance principle was stated by bramson cox and le gall as a consequence of a theorem of cox durrett and perkins less precise estimates are derived in dimension d 4

the geometry of a parameter space of interacting particle systems we study a four parameter family of interacting particle systems containing the basic voter model and contact processes two processes in this family are related by duality or thinning if and only if their parameters belong to the same orbit of a certain one dimensional group of linear mappings this shows that many duals exist

music type electromagnetic imaging of a collection of small three dimensional inclusions in this paper we consider the localization of a collection of small three dimensional bounded homogeneous inclusions via time harmonic electromagnetic means typically using arrays of electric or magnetic dipole transmitters and receivers with given polarization s at some distance from the collection possibly also lying in the far field the inclusions somewhat apart or closely spaced are buried within a homogeneous medium and are of arbitrary contrast of permittivity conductivity and permeability vis ag vis this embedding medium the problem is formulated as an inverse scattering problem for the full maxwell equations and it involves a robust asymptotic modeling of the multistatic response matrix no specific application is studied at this stage but characterization of obstacles in subsoils nondestructive evaluation of man made structures and medical imaging are primary fields of application envisaged the proposed approach uses a music multiple signal classification type algorithm and it yields fast numbering accurate localization and estimates of the electromagnetic and geometric parameters polarization tensors of the inclusions the mathematical machinery is detailed first some specific attention being given to triaxial ellipsoidal inclusions and degenerate spherical shapes for the latter known results are retrieved then the viability of this algorithm which would be easily extended to planarly layered environments by introduction of their green s functions is documented by a variety of numerical results from synthetic noiseless and severely noisy field data

an asymptotic factorization method for inverse electromagnetic scattering in layered media we consider the inverse problem to reconstruct the number and the positions of a collection of finitely many small perfectly conducting scatterers buried within the lower halfspace of an unbounded two layered background medium from near field measurements of time harmonic electromagnetic fields for this purpose we first study the direct scattering problem and derive an asymptotic expansion of the scattered field as the size of the scatterers tends to zero integral equation methods and a factorization of the corresponding near field measurement operator are applied to prove this expansion in the second part of this work we use the asymptotic expansion to justify a noniterative reconstruction algorithm which is a combination of factorization methods and music type methods we illustrate the feasibility of this method by a numerical example

stochastic p d e s arising from the long range contact and long range voter processes a long range contact process and a long range voter process are scaled so that the distance between sites decreases and the number of neighbors of each site increases the approximate densities of occupied sites under suitable tine scaling converge to continuous space time densities which solve stochastic p d e s for the contact process the limiting equation is the kolmogorov petrovskii piscuinov equation driven by branching white noise for the voter process the limiting equation is the heat equation driven by fisher wright white noise

vortices and strings in a model ecosystem we study the spatial pattern formation in a model ecosystem by the position fixed reaction method this ecosystem contains three biospecies whose competing powers are cyclic it is well known that this system is self organized into a quasistationary state and that the mean field approximation mfa never predicts such a pattern formation recently several authors applied the pair approximation pa and obtained considerable improvements however applying pa to our ecosystem fails to yield such an improvement as revealed by computer simulations the failures of mfa and pa may be attributed to the fact that both approximations neglect a long range correlation thus we introduce the concept of topological defects such as vortices or strings and demonstrate that the dynamics of these defects can at least qualitatively account for the observed pattern formation dynamics

a direct impedance tomography algorithm for locating small inhomogeneities impedance tomography seeks to recover the electrical conductivity distribution inside a body from measurements of current flows and voltages on its surface in its most general form impedance tomography is quite ill posed but when additional a priori information is admitted the situation changes dramatically in this paper we consider the case where the goal is to find a number of small objects inhomogeneities inside an otherwise known conductor taking advantage of the smallness of the inhomogeneities we can use asymptotic analysis to design a direct i e non iterative reconstruction algorithm for the determination of their locations the viability of this direct approach is documented by numerical examples

exponential decay for subcritical contact and percolation processes we study the contact process together with a version of the percolation process with one continuously varying coordinate it is proved here that the radius of the infected cluster has an exponentially decaying tail throughout the subcritical phase the same is true of the lebesgue measure in space time of this cluster certain critical exponent inequalities are derived and the critical point of the percolation process in two dimensions is determined exactly

clustering in the one dimensional three color cyclic cellular automaton cyclic cellular automaton the author has previously shown that this process fluctuates meaning that each lattice site changes color infinitely often so that there is no final state for the system the focus of the current work is on the clustering properties of this system this paper demonstrates that the one dimensional three color cyclic cellular automaton clusters and the mean cluster size as a function of time t is asymptotic to ct1 2 where c is an explicitly calculable constant the method of proof also allows us to compute asymptotic estimates of the mean interparticle distance for a one dimensional system of particles which undergo deterministic motion and which annihilate upon collision no clustering results are known about the four color process but evidence is presented to suggest that the mean cluster size of such systems grows at a rate different from t1 2

the brownian web characterization and convergence the brownian web bw is the random network formally consist ing of the paths of coalescing one dimensional brownian motions start ing from every space time point in r r we extend the earlier work of arratia and of toth and werner by providing characterization and convergence results for the bw distribution including convergence of the system of all coalescing random walks to the bw under diffusive space time scaling we also provide characterization and convergence results for the double brownian web which combines the bw with its dual process of coalescing brownian motions moving backwards in time with forward and backward paths reflecting off each other for the bw deterministic space time points are almost surely of type 0 1 zero paths into the point from the past and exactly one path out of the point to the future we determine the hausdorff dimension for all types that actually occur dimension 2 for type 0 1 3 2 for 1 1 and 0 2 1 for 1 2 and 0 for 2 1 and 0 3

global divergence of spatial coalescents we study several fundamental properties of a class of stochastic processes called spatial coalescents in these models a number of particles perform independent random walks on some underlying graph g in addition particles on the same vertex merge randomly according to a given coalescing mechanism a remarkable property of mean field coalescent processes is that they may come down from infinity meaning that starting with an infinite number of particles only a finite number remains after any positive amount of time almost surely we show here however that in the spatial setting on any infinite and bounded degree graph the total number of particles will always remain infinite at all times almost surely moreover if g mathbb z d and the coalescing mechanism is kingman s coalescent then starting with n particles at the origin the total number of particles remaining is of order log n d at any fixed positive time where log is the inverse tower function at sufficiently large times the total number of particles is of order log n d 2 when d 2 we provide parallel results in the recurrent case d 2 the spatial beta coalescents behave similarly where log log n is replacing log n

the dynamics of defect ensembles in one dimensional cellular automata we investigate the dynamics of ensembles of diffusive defects in one dimensional deterministic cellular automata the work builds on earlier results on individual random walks in cellular automata here we give a natural condition guaranteeing diffusive behavior also in the presence of other defects simple branching and birth mechanisms are introduced and prototype classes of cellular automata exhibiting weakly interacting walks capable of annihilation and coalescence are studied their equilibrium behavior is also characterized the design principles of cellular automata with desired diffusive interaction properties become transparent from this analysis

solution to urn models of pairwise interaction with application to social physical and biological sciences we investigate a family of urn models that correspond to one dimensional random walks with quadratic transition probabilities that have highly diverse applications well known instances of these two urn models are the ehrenfest model of molecular diffusion the voter model of social influence and the moran model of population genetics we also provide a generating function method for diagonalizing the corresponding transition matrix that is valid if and only if the underlying mean density satisfies a linear differential equation and express the eigenvector components as terms of ordinary hypergeometric functions the nature of the models lead to a natural extension to interaction between agents in a general network topology we analyze the dynamics on uncorrelated heterogeneous degree sequence networks and relate the convergence times to the moments of the degree sequences for various pairwise interaction mechanisms

self intersections of 1 dimensional random walks consider a random walk snon the integers where the steps i have mean 0 and variance 2 let t be the time of first self intersection of the random walk it is shown that as t grows at rate 2 3 more precisely t 2 3 has a non degenerate limit distribution which can be described in terms of brownian motion local time

measure preserving words are primitive a word w 2 fk the free group on k generators is called primitive if it belongs to some basis of fk associated with w and a finite group g is the word map w g g g defined on the direct product of k copies of g we call w measure preserving if given uniform measure on g g the image of this word map induces uniform measure on g for every finite group g it is easy to see that every primitive word is measure preserving and several authors have conjectured that the two properties are in fact equivalent here we prove this conjecture the main ingredients of the proof include random coverings of stallings graphs algebraic extensions of free groups and mobius inversions our methods yield the stronger result that a subgroup of fk is measure preserving iff it is a free factor as an interesting corollary of this result we resolve a question on the profinite topology of free groups and show that the primitive elements of fk form a closed set in this topology

identification of small inhomogeneities of extreme conductivity by boundary measurements a theorem on continuous dependence we consider an electrostatic problem for a conductor consisting of finitely many small inhomogeneities of extreme conductivity embedded in a spatially varying reference medium firstly we establish an asymptotic formula for the voltage potential in terms of the reference voltage potential the location of the inhomogeneities and their geometry secondly we use this representation formula to prove a lipschitz continuous dependence estimate for the corresponding inverse problem this estimate bounds the difference in the location and in certain geometric properties of two sets of inhomogeneities by the difference in the boundary voltage potentials corresponding to a fixed current distribution

super brownian limits of voter model clusters the voter model is one of the standard interacting particle systems two related problems for this process are to analyze its behavior after large times t for the sets of sites 1 sharing the same opinion as the site 0 and 2 having the opinion that was originally at 0 results on the sizes of these sets were given by sawyer 1979 and bramson and griffeath 1980 here we investigate the spatial structure of these sets in d 2 which we show converge to quantities associated with super brownian motion after suitable normalization the main theorem from cox durrett and perkins 2000 serves as an important tool for these results

stochastic spatial models in the models we will consider space is represented by a grid of sites that can be in one of a finite number of states and that change at rates that depend on the states of a finite number of sites our main aim here is to explain an idea of durrett and levin 1994 the behavior of these models can be predicted from the properties of the mean field ode i e the equations for the densities of the various types that result from pretending that all sites are always independent we will illustrate this picture through a discussion of eight families of examples from statistical mechanics genetics population biology epidemiology and ecology some of our findings are only conjectures based on simulation but in a number of cases we are able to prove results for systems with fast stirring by exploiting connections between the spatial model and an associated reaction diffusion equation

diffusive clustering in the two dimensional voter model on donne des resultats de simulation et une enquete concernant les resultats connus sur le modele du votant dans trois cas distincts trois ou plusieurs dimensions une dimension et deux dimensions

ergodic theorems for the multitype contact process this paper studies a process involving competition of two types of particles 1 and 2 for the empty space 0 each site of the latticez d is therefore in one of three possible states 0 1 or 2 particles of each type die with rate 1 while an empty site becomes occupied by a particle of typei with rate i proportion of neighbors of typei the set of neighbors of a sitex is of the form y x y j for a positive integerj and a norm assuming there are only 0 s and 1 s present at the beginning the process reduces to the contact process with the critical rate of survival of 1 s being c the basic problem we address is the existence of equilibria in which both types of particles coexist without loss of generality one can restrict to the case 2 1 c and in this case we show 1 if 2 1 and the initial state is translation invariant and contains infinitely many 2 s then the 1 s go away and the process approaches the invariant measure of the contact process with only 2 s and 0 s present 2 if 2 1 andd 2 then clustering occurs starting from a translation invariant initial measure with no mass on all 0 s the process converges weakly to a convex combination of the two invariant measures obtained with only one type of particles present and 3 if 2 1 andd 3 then there is a one parameter family of invariant measures including both types

recovery of an embedded obstacle and the surrounding medium for maxwell s system abstract in this paper we are concerned with the inverse electromagnetic scattering problem of recovering a complex scatterer by the corresponding electric far field data the complex scatterer consists of an inhomogeneous medium and a possibly embedded perfectly electric conducting pec obstacle the far field data are collected corresponding to incident plane waves with a fixed incident direction and a fixed polarisation but frequencies from an open interval it is shown that the embedded obstacle can be uniquely recovered by the aforementioned far field data independent of the surrounding medium furthermore if the surrounding medium is piecewise homogeneous then the medium can be recovered as well those unique recovery results are new to the literature our argument is based on low frequency expansions of the electromagnetic fields and certain harmonic analysis techniques

consolidation rates for two interacting systems in the plane this paper is a sequel of a paper of cox and griffeath diffusive clustering in the two dimensional voter model we continue our study of the voter model and coalescing random walks on the two dimensional integer lattice some exact asymptotics concerning the rate of clustering in the former process and the coalescence rate of the latter are derived we use these results to prove a limit law announced in that earlier paper concerning the size of the largest square centered at the origin which is of solid color at a large time t

a phase transition for a stochastic pde related to the contact process we consider the one dimensional heat equation with a semilinear term and with a nonlinear white noise term r durrett conjectured that this equation arises as a weak limit of the contact process with longrange interactions we show that our equation possesses a phase transition to be more precise we assume that the initial function is nonnegative with bounded total mass if a certain parameter in the equation is small enough then the solution dies out to 0 in finite time with probability 1 if this parameter is large enough then the solution has a positive probability of never dying out to 0 this result answers a question of durett

mathematical analysis of plasmonic resonances for nanoparticles the full maxwell equations abstract in this paper we use the full maxwell equations for light propagation in order to analyze plasmonic resonances for nanoparticles we mathematically define the notion of plasmonic resonance and analyze its shift and broadening with respect to changes in size shape and arrangement of the nanoparticles using the layer potential techniques associated with the full maxwell equations we present an effective medium theory for resonant plasmonic systems and derive a condition on the volume fraction under which the maxwell garnett theory is valid at plasmonic resonances

asymptotic behavior of densities for two particle annihilating random walks consider the system of particles on d where particles are of two types a andb and execute simple random walks in continuous time particles do not interact with their own type but when ana particle meets ab particle both disappear i e are annihilated this system serves as a model for the chemical reactiona b inert we analyze the limiting behavior of the densities a t and b t when the initial state is given by homogeneous poisson random fields we prove that for equal initial densities a 0 b 0 there is a change in behavior fromd 4 where a t b t c t d 4 tod 4 where a t b t c tast for unequal initial densities a 0 b 0 a t e c l ind 1 a t e ct logt ind 2 and a t e ct ind 3 the termc depends on the initial densities and changes withd techniques are from interacting particle systems the behavior for this two particle annihilation process has similarities to those for coalescing random walks a a a and annihilating random walks a a inert the analysis of the present process is made considerably more difficult by the lack of comparison with an attractive particle system

multistatic response matrix of a 3 d inclusion in half space and music imaging recent work on the retrieval of 3 d bounded dielectric and or magnetic inclusions in free space is extended to burial in a half space though emphasis is on the case of a single inclusion enabling the derivation of closed form mathematical results in illuminating fashion the approach extends to the case of an unknown number of well separated inclusions within an asymptotic field formulation derived from exact contrast source vector integral formulations satisfied by the time harmonic fields and using proper reciprocity relationships of the dyadic green s functions the multistatic response matrix of the inclusion is constructed from the leading order term of the fields its singular value structure is analyzed in detail for a dielectric or a magnetic contrast or both contrasts this is performed in the case of a unique electric dipole array operated in the transmit receive mode at a single frequency a multiple signal classification type algorithm follows from the decomposition yielding a cost functional the magnitude of which peaks at the inclusion center numerical results illustrate the above as a function of the geometric and electromagnetic parameters of the configuration imaging of a spherical inclusion is in particular investigated from severely noisy synthetic data as well as of two inclusions embedded within a non conductive or conductive half space

self organized critical forest fire model a forest fire model is introduced which contains a lightning probability f this leads to a self organized critical state in the limit f ensuremath rightarrow 0 provided that the time scales of tree growth and burning down of forest clusters are separated we derive scaling laws and calculate all critical exponents the values of the critical exponents are confirmed by computer simulations for a two dimensional system we show that the forest density in the critical state assumes its minimum possible value i e that energy dissipation is maximum

relative pro l completions of mapping class groups abstract fix a prime number l in this paper we develop the theory of relative pro l completion of discrete and profinite groups a natural generalization of the classical notion of pro l completion and show that the pro l completion of the torelli group does not inject into the relative pro l completion of the corresponding mapping class group when the genus is at least 2 see theorem 1 below as an application we prove that when g 2 the action of the pro l completion of the torelli group t g 1 on the pro l fundamental group of a pointed genus g surface is not faithful the choice of a first order deformation of a maximally degenerate stable curve of genus g determines an action of the absolute galois group g q on the relative pro l completion of the corresponding mapping class group we prove that for all g all such representations are unramified at all primes l when the first order deformation is suitably chosen this proof was communicated to us by mochizuki and tamagawa

congruence topologies on the mapping class group abstract let s be the pure mapping class group of a connected orientable surface s of negative euler characteristic for c a class of finite groups let 1 s c be the pro c completion of the fundamental group of s the c congruence completion s c of s is the profinite completion induced by the embedding s out 1 s c in this paper we begin a systematic study of such completions for different c we show that the combinatorial structure of the profinite groups s c closely resemble that of s a fundamental question is how c congruence completions compare with pro c completions even though in general e g for c the class of finite solvable groups s c is not even virtually a pro c group we show that for z 2 c g s 2 and s open there is a natural epimorphism from the c congruence completion s 2 c of the abelian level of order 2 to its pro c completion s 2 c in particular this is an isomorphism for the class of finite groups and for the class of 2 groups moreover in these two cases the result also holds for a closed surface

coexistence results for catalysts in this paper we consider a modification of ziff gulari and barshad s 1986 model of oxidation of carbon monoxide on a catalyst surface in which the reactants are mobile on the catalyst surface we find regions in the parameter space in which poisoning occurs the catalyst surface becomes completely occupied by one type of atom and another in which there is a translation invariant stationary distribution in which the two atoms have positive density the last result is proved by exploiting a connection between the particle system with fast stirring and a limiting system of reaction diffusion equations

improved upper bounds for the contact process critical value the best known upper bound for the critical value c of the basic one dimensional contact process is 2 most techniques for finding bounds on critical values have the property that they can be modified in order to obtain improved bounds this seemed not to be the case for the approach which yielded c 2 for the basic contact process in this paper we propose a technique for generating better bounds in this context to illustrate its use we carry out the full program in one case with the conclusion that c 1 942

a spatial model for the abundance of species the voter model with mutations occurring at a positive rate has a unique equilibrium distribution we investigate the logarithms of the relative abundance of species for these distributions in d 2 we show that as 0 the limiting distribution is right triangular in d 2 and uniform in d 3 we also obtain more detailed results for the histograms that biologists use to estimate the underlying density functions

the discrete evolution model of bak and sneppen is conjugate to the classical contact process two fundamental models of critical phenomena are connected we show that the discrete bak sneppen evolution model is conjugate to the classical contact process this holds in discrete and continuous time on all graphs and for random as well as for deterministic choice of neighbors thus the extensive theory for the contact process applies to the discrete bak sneppen model too

on finitely generated profinite groups i strong completeness and uniform bounds we prove that in every finitely generated profinite group every subgroup of finite index is open this implies that the topology on such groups is determined by the algebraic structure this is deduced from the main result about finite groups let w be a locally finite group word and d n then there exists f f w d such that in every d generator finite group g every element of the verbal subgroup w g is equal to a product of f w values an analogous theorem is proved for commutators this implies that in every finitely generated profinite group each term of the lower central series is closed the proofs rely on some properties of the finite simple groups to be established in part ii

the critical contact process dies out by producing a finite box criterion for survival in a slab we show that the critical contact process dies out and that in two and more dimensions the critical parameter value coincides with the limit of slab thresholds using the techniques developed in this paper one may obtain the complete convergence and shape theorems for contact processes in all dimensions and for all infection rates our results apply also to the discrete time analogue of the contact process viz oriented percolation

contact processes in several dimensions this paper deals primarily with the basic contact processes introduced and studied by harris 12t4 these are random evolutions on the state space s a l l subsets of z a zd the d dimensional integer lattice with extremely simple local dynamics namely if we think of the contact process as representing the spread of an infection t es denoting the set of infected sites at time t then x e it becomes healthy at exponential rate 1 while x z a t is infected at a rate proport ional to the number of sites neighboring x where infection is present the proportionali ty constant 2 is called the infection parameter contact processes are perhaps the simplest s valued markov processes which exhibit a critical phenomenon infection emanating from a single site dies out with probabili ty one for small positive 2 but has positive probability of surviving for all time when is large there is a critical value 2 d where the transition occurs principal objectives of analysis are the precise formulation of the critical phenomenon and detailed description of the ergodic behavior both below and above a c our recent survey articles 10 and i 6 provide introductions to contact processes and to the general field of interacting s valued systems respectively we will make constant use of notation and techniques from the surveys assuming that the reader is familiar with thern the article 10 describes in some detail the current state of knowledge concerning one dimensional contact processes the theory for d 1 is now fairly complete with two major exceptions 1 is not even determined to one decimal place 1 18 2 e there is an

identification of small inhomogeneities asymptotic factorization we consider the boundary value problem of calculating the electrostatic potential for a homogeneous conductor containing finitely many small insulating inclusions we give a new proof of the asymptotic expansion of the electrostatic potential in terms of the background potential the location of the inhomogeneities and their geometry as the size of the inhomogeneities tends to zero such asymptotic expansions have already been used to design direct i e noniterative reconstruction algorithms for the determination of the location of the small inclusions from electrostatic measurements on the boundary e g music type methods our derivation of the asymptotic formulas is based on integral equation methods it demonstrates the strong relation between factorization methods and music type methods for the solution of this inverse problem

asymptotic density in a threshold coalescing and annihilating random walk we consider an interacting random walk on z d where particles interact only at times when a particle jumps to a site at which there are at least n 1 other particles present if there are i n 1 particles present then the particle coalesces is removed from the system with probability c i and annihilates is removed along with another particle with probability a i we call this process the n threshold randomly coalescing and annihilating random walk we show that for n 3 if both a i and a i c i are increasing in i and if the dimension d is at least 2n 4 then p the origin is occupied at time t c d n t 1 n 1 e number of particles at the origin at time t c d n t 1 the constants c d n are explicitly identified the proof is an extension of a result obtained by kesten and van den berg for the 2 threshold coalescing random walk and is based on an approximation for de t dt