complete convergence theorem for a competition model in this paper we consider a hierarchical competition model durrett and swindle have given sufficient conditions for the existence of a nontrivial stationary distribution here we show that under a slightly stronger condition the complete convergence theorem holds and hence there is a unique nontrivial stationary distribution

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 survival probability for critical spread out oriented percolation above 4 1 dimensions i induction we consider critical spread out oriented percolation above 4 1 dimensions our main result is that the extinction probability at time n i e the probability for the origin to be connected to the hyperplane at time n but not to the hyperplane at time n 1 decays like 1 bn 2 as n to infty where b is a finite positive constant this in turn implies that the survival probability at time n i e the probability that the origin is connected to the hyperplane at time n decays like 1 bn as n to infty the latter has been shown in an earlier paper to have consequences for the geometry of large critical clusters and for the incipient infinite cluster the present paper is part i in a series of two papers in part ii we derive a lace expansion for the survival probability adapted so as to deal with point to plane connections this lace expansion leads to a nonlinear recursion relation for the survival probability in part i we use this recursion relation to deduce the asymptotics via induction

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

percolation in half spaces equality of critical densities and continuity of the percolation probability renormalization arguments are developed and applied to independent nearest neighbor percolation on various subsets of d d 2 yielding equality of the critical densities p c for a half space quarter space etc and ford 2 equality with the limit of slab critical densities continuity of the phase transition for the half space quarter space etc i e vanishing of the percolation probability p atp p c

time dependent perturbation theory for nonequilibrium lattice models we develop a time dependent perturbation theory for nonequilibrium interacting particle systems we focus on models such as the contact process which evolve via destruction and autocatalytic creation of particles at a critical value of the destruction rate there is a continuous phase transition between an active steady state and the vacuum state which is absorbing we present several methods for deriving series for the evolution starting from a single seed particle including expansions for the ultimate survival probability in the super and subcritical regions expansions for the average number of particles in the subcritical region and short time expansions algorithms for computer generation of the various expansions are presented rather long series 24 terms or more and precise estimates of critical parameters are presented

hyperscaling inequalities for the contact process and oriented percolation the contact process and oriented percolation are expected to exhibit the same critical behavior in any dimension above their upper critical dimension dc they exhibit the same critical behavior as the branching process assuming existence of the critical exponents we prove a pair of hyperscaling inequalities which together with the results of refs 16 and 18 implies dc 4

the box crossing property for critical two dimensional oriented percolation we consider critical oriented bernoulli percolation on the square lattice mathbb z 2 we prove a russo seymour welsh type result which allows us to derive several new results concerning the critical behavior the sub linear polynomial fluctuations contrast with the supercritical regime where w n is known to behave linearly in n it is also different from the critical picture obtained for non oriented bernoulli percolation in which the scaling limit is non degenerate in both directions all our results extend to the graphical representation of the one dimensional contact process

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

a contact process with a single inhomogeneous site the one dimensional basic contact process is a markov process for which particles give birth on vacant nearest neighbor sites at rate 0 and particles die at rate one we introduce a one dimensional contact process with a single inhomogeneous site the evolution is as above except that a particle located at the origin does not die let c be the critical value of the basic contact process we show that for c the upper invariant measures of the inhomogeneous contact process and the basic contact process coincide except at a finite number of sites the behavior at c is much more intersting the upper invariant measure of the inhomogeneous contact process concentrates on configurations with infinitely many particles while it is known that the critical basic contact process dies out so a single inhomogeneity may provoke a perturbation unbounded in space as a byproduct of our analysis we prove that the connectivity probabilities of the critical basic contact process are not summable we also give a biological interpretation of this model

percolation and contact processes with low dimensional inhomogeneity we consider inhomogeneous nearest neighbor bernoulli bond percolation on z d where the bonds in a fixed s dimensional hyperplane 1 s d 1 have density p 1 and all other bonds have fixed density p c z d the homogeneous percolation critical value for s 2 it is natural to conjecture that there is a new critical value p c s z d for p 1 strictly between p c z d and p c z s we prove this for large d and 2 s d 3 for s 1 it is natural to conjecture that p c 1 z d 1 as shown for d 2 by zhang we prove this for large d related results for the contact process are also presented

existence of a constant for finite system extinction we show the existence of a constant 0 such that if n is the extinction time for a supercritical contact process on 1 n d starting from full occupancy then log e n nd tend to as n tends to infinity

the complete convergence theorem holds for contact processes on open clusters of ℤd ℤ we study contact processes on open clusters of half space our result shows that the complete convergence theorem holds

amplifiers for the moran process the moran process as studied by lieberman hauert and nowak is a randomised algorithm modelling the spread of genetic mutations in populations the algorithm runs on an underlying graph where individuals correspond to vertices initially one vertex chosen uniformly at random possesses a mutation with fitness r 1 all other individuals have fitness 1 during each step of the algorithm an individual is chosen with probability proportional to its fitness and its state mutant or nonmutant is passed on to an out neighbour which is chosen uniformly at random if the underlying graph is strongly connected then the algorithm will eventually reach fixation in which all individuals are mutants or extinction in which no individuals are mutants an infinite family of directed graphs is said to be strongly amplifying if for every r 1 the extinction probability tends to 0 as the number of vertices increases a formal definition is provided in the article strong amplification is a rather surprising property it means that in such graphs the fixation probability of a uniformly placed initial mutant tends to 1 even though the initial mutant only has a fixed selective advantage of r 1 independently of n the name strongly amplifying comes from the fact that this selective advantage is amplified strong amplifiers have received quite a bit of attention and lieberman et al proposed two potentially strongly amplifying families superstars and metafunnels heuristic arguments have been published arguing that there are infinite families of superstars that are strongly amplifying the same has been claimed for metafunnels in this article we give the first rigorous proof that there is an infinite family of directed graphs that is strongly amplifying we call the graphs in the family megastars when the algorithm is run on an n vertex graph in this family starting with a uniformly chosen mutant the extinction probability is roughly n 1 2 up to logarithmic factors we prove that all infinite families of superstars and metafunnels have larger extinction probabilities as a function of n finally we prove that our analysis of megastars is fairly tight there is no infinite family of megastars such that the moran algorithm gives a smaller extinction probability up to logarithmic factors also we provide a counterexample which clarifies the literature concerning the isothermal theorem of lieberman et al

critical behavior and the limit distribution for long range oriented percolation i we consider oriented percolation on mathbb z d times mathbb z whose bond occupation probability is pd where p is the percolation parameter and d is a probability distribution on mathbb z d suppose that d x decays as x d for some 0 we prove that the two point function obeys an infrared bound which implies that various critical exponents take on their respective mean field values above the upper critical dimension d c 2 alpha wedge2 we also show that for every k the fourier transform of the normalized two point function at time n with a proper spatial scaling has a convergent subsequence to e c k alpha wedge2 for some c 0

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

gaussian limit for critical oriented percolation in high dimensions in this paper we consider the spread out oriented bond percolation models inz d z withd 4 and the nearest neighbor oriented bond percolation model in sufficiently high dimensions let n n 1 2 be the random measures defined onr d by eta n a sum limits x in z d 1 a x sqrt n 1 0 0 to x n the mean of n denoted by bar eta n is the measure defined by bar eta n a e p eta n a we use the lace expansion method to show that the sequence of probability measures bar eta n r d 1 bar eta n converges weakly to a gaussian limit asn for everyp in the subcritical regime as well as the critical regime of these percolation models also we show that for these models the parallel correlation length xi p p c p 1 asp pc

mean field critical behavior for the contact process the contact process is a model of spread of an infectious disease combining with the result of ref 1 we prove that the critical exponents take on the mean field values for sufficiently high dimensional nearest neighbor models and for sufficiently spread out models with d 4 c as c and c 1 as c where and are the spread probability and the susceptibility of the infection respectively and c is the critical infection rate our results imply that the upper critical dimension for the contact process is at most 4

phase transitions in the quadratic contact process on complex networks the quadratic contact process qcp is a natural extension of the well studied linear contact process where infected 1 individuals infect susceptible 0 neighbors at rate ensuremath lambda and infected individuals recover 1 0 at rate 1 in the qcp a combination of two 1 s is required to effect a 0 1 change we extend the study of the qcp which so far has been limited to lattices to complex networks we define two versions of the qcp vertex centered vqcp and edge centered eqcp with birth events 1 ensuremath 0 ensuremath 1 1 ensuremath 1 ensuremath 1 and 1 ensuremath 1 ensuremath 0 1 ensuremath 1 ensuremath 1 respectively where ensuremath represents an edge we investigate the effects of network topology by considering the qcp on random regular erd ifmmode mbox h o else h o fi s r enyi and power law random graphs we perform mean field calculations as well as simulations to find the steady state fraction of occupied vertices as a function of the birth rate we find that on the random regular and erd ifmmode mbox h o else h o fi s r enyi graphs there is a discontinuous phase transition with a region of bistability whereas on the heavy tailed power law graph the transition is continuous the critical birth rate is found to be positive in the former but zero in the latter

random walk on the incipient infinite cluster for oriented percolation in high dimensions we consider simple random walk on the incipient infinite cluster for the spread out model of oriented percolation on mathbb z d times mathbb z in dimensions d 6 we obtain bounds on exit times transition probabilities and the range of the random walk which establish that the spectral dimension of the incipient infinite cluster is frac 4 3 and thereby prove a version of the alexander orbach conjecture in this setting the proof divides into two parts one part establishes general estimates for simple random walk on an arbitrary infinite random graph given suitable bounds on volume and effective resistance for the random graph a second part then provides these bounds on volume and effective resistance for the incipient infinite cluster in dimensions d 6 by extending results about critical oriented percolation obtained previously via the lace expansion

the contact process on trees the contact process on an infinite homogeneous tree is shown to exhibit at least two phase transitions as the infection parameter is varied for small values of a single infection eventually dies out for larger the infection lives forever with positive probability but eventually leaves any finite set the survival probability is a continuous function of and the proof of this is much easier than it is for the contact process on d dimensional integer lattices for still larger the infection converges in distribution to a nontrivial invariant measure for an n ary tree with n large the first of these transitions occurs when 1 n and the second occurs when 1 2 n e n nonhomogeneous trees whose vertices have degrees varying between 1 and n behave essentially as homogeneous n ary trees provided that vertices of degree n are not too rare in particular letting n go to galton watson trees whose vertices have degree n with probability that does not decrease exponentially with n may have both phase transitions occur together at 0 the nature of the second phase transition is not yet clear and several problems are mentioned in this regard

asymptotically optimal amplifiers for the moran process abstract we study the moran process as adapted by lieberman hauert and nowak this is a model of an evolving population on a graph or digraph where certain individuals called mutants have fitness r and other individuals called non mutants have fitness 1 we focus on the situation where the mutation is advantageous in the sense that r 1 a family of digraphs is said to be strongly amplifying if the extinction probability tends to 0 when the moran process is run on digraphs in this family the most amplifying known family of digraphs is the family of megastars of galanis et al we show that this family is optimal up to logarithmic factors since every strongly connected n vertex digraph has extinction probability n 1 2 next we show that there is an infinite family of undirected graphs called dense incubators whose extinction probability is o n 1 3 we show that this is optimal up to constant factors finally we introduce sparse incubators for varying edge density and show that the extinction probability of these graphs is o n m where m is the number of edges again we show that this is optimal up to constant factors

t e harris contributions to interacting particle systems and percolation this is a bird s eye view of t e harris work on interacting particle systems and percolation and of its impact on later work in probability theory and mathematical physics nal illness and death was a great loss to me personally and professionally as well as to probability theory in southern california and beyond for three decades he and i were the primary probabilists at usc and ucla respectively our approach to mathematics and our mathematical interests were very similar my wife chris and i enjoyed many wonderful social occasions at the harris home in beverly hills with ted and connie they were great hosts the conversation was always stimulating they were interested in so many things ted initiated the southern california probability symposium in about 1970 it is probably the oldest meeting of its type in the us and continues to provide an exciting place for interactions among southern california probabilists to this day in two of the years 1989 and 2007 the meeting was dedicated to ted the rst was on the occasion of his 70th birthday and the second was in his memory even after his retirement ted attended the usc probability seminar regularly always sitting in the front row and asking perceptive questions few mathematicians have had a greater ratio of number of ideas to number of papers ted wrote many fewer papers 31 than most prominent mathematicians but each is a jewel in each of the several areas in which he worked he was among the

interface motion in random media we study the pinning phase transition for discrete surface dynamics in random environments a renormalization procedure is devised to prove that the interface moves with positive velocity under a finite size condition this condition is then checked for different examples of microscopic dynamics to illustrate the flexibility of the method we show in our examples the existence of a phase transition for various models including high dimensional interfaces dependent environments and environments with arbitrarily deep obstacles finally our ballisticity criterion is proved to be valid up to the critical threshold for a lipschitz interface model

isotropic growth on a grid abstract isotropic growth from a single point on a two dimensional square grid should generate an increasing sequence of discretized discs we present a simple probabilistic model for growth on a grid and discuss a class of parameterizations of the model called kernels which was conjectured s thompson and a rosenfeld discrete stochastic growth models for two dimensional shapes in shape in picture mathematical descriptions of shape in grey level images y l o a toet d h foster and p meer eds 301 318 springer verlag heidelberg 1993 to produce isotropic growth we disprove this conjecture but we claim that these kernels produce growth that can be decomposed into isotropic and nonisotropic other probabilistic growth processes on grids and describe qualitative and quantitative properties of the models we also consider a deterministic growth model based on the diffusion equation and show empirically that discretization of this model leads to a steady state configuration that appears to be polygonal

the threshold contact process a continuum limit in the threshold contact process on thed dimensional integer lattice with ranger healthy sites become infected at rate if they have at least one infectedr neighbour and recover at rate 1 we show that the critical value c r is asymptotic tor d c asr where c is the critical value of the birth rate for a continuum threshold contact process which may be described in terms of an oriented continuous percolation model driven by a poisson process of rate ind 1 dimensions we have bounds of 0 7320 c 3 ford 1

the survival of nonattractive interacting particle systems on z we consider interacting particle systems on z which allow five types of pairwise interaction annihilation birth coalescence death and exclusion with corresponding rates a b c d e we show that whatever the values of a c d e if the birthrate is high enough there is a positive probability the particle system will survive starting from any finite occupied set in particular an ips with rates a b c d e has a positive probability of survival if b 4d 6a c a d e or b 7d 3a 3c 3e c a d e we create a suitable supermartingale by extending the method used by holley and liggett in their treatment of the contact process

the second lowest extremal invariant measure of the contact process ii we continue the investigation of the behavior of the contact process on infinite connected graphs of bounded degree some questions left open by salzano and schonmann 1997 concerning the notions of complete convergence partial convergence and the criterion r s are answered the continuity properties of the survival probability and the recurrence probability are studied these order parameters are found to have a richer behavior than expected with the possibility of the survival probability being discontinuous at or above the threshold for survival a condition which guarantees the continuity of the survival probability above the survival point is introduced and exploited the recurrence probability is shown to always be left continuous above the recurrence point and a necessary and sufficient condition for its right continuity is introduced and exploited it is shown that for homogeneous graphs the survival probability can only be discontinuous at the survival point and the recurrence probability can only be discontinuous at the recurrence point for graphs which are obtained by joining a finite number of severed homogeneous trees by means of a finite number of vertices and edges the survival point the recurrence point and the discontinuity points of the survival and recurrence probabilities are located

a spatial stochastic model for rumor transmission we consider an interacting particle system representing the spread of a rumor by agents on the d dimensional integer lattice each agent may be in any of the three states belonging to the set 0 1 2 here 0 stands for ignorants 1 for spreaders and 2 for stiflers a spreader tells the rumor to any of its nearest ignorant neighbors at rate at rate a spreader becomes a stifler due to the action of other nearest neighbor spreaders finally spreaders and stiflers forget the rumor at rate one we study sufficient conditions under which the rumor either becomes extinct or survives with positive probability

large deviations and ergodicity for spin particle systems in this paper we investigate the large deviation principle ldp for spin particle systems with possibly vanishing flip rates the situation turns out to be much more complicated if the flip rates are allowed to be zero than the one considered by dai where the systems are assumed to have strictly positive flip rates the upper and lower large deviation bounds are studied respectively the two governing rate functions are compared and a variational principle is given we then apply the results to obtain some new large deviation estimates for the occupation times of attractive systems in particular we prove a strong form of exponential convergence for ergodic systems

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

on the behavior of some cellular automata related to bootstrap percolation convergence to total occupancy and irc the threshold for this convergence to occur exponentially fast we locate these critical points for all the bootstrap percolation models showing that they are both 0 when 1 d for certain rules in which the orientation is important we show that 0 pc ir 1 by relating these systems to oriented site percolation finally these oriented models are used to obtain an estimate for a critical exponent of these models 1 introduction the fields of interacting particle systems mathematical statistical mechanics and percolation have benefited very much from their interrelations here we study a family of models which has arisen in the interface among these areas cellular automata such as those studied in this article may be considered as interacting particle systems see liggett 1985 for a survey of this field the relations between the models that we consider and percolation will become clear in many of the proofs given but can already be guessed from the fact that some of these systems are known as bootstrap percolation finally relations with statistical mechanics while not so explicit in this article were clearly present for instance in the article by chalupa leath and reich 1979 where bootstrap percolation was introduced in connection to disordered magnetic systems also in aizenman and lebowitz 1988 the motivation for studying these systems came from the nonequilibrium statistical mechanics problem of metastability our main concern in this article will be with the critical behavior of our models that is how their behavior changes qualitatively as some parameters cross certain values critical points as usual one of the main tools in the analysis of such phenomena will be a sort of renormalization procedure by

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

information transmission and criticality in the contact process in the present paper we study the relation between criticality and information transmission in the one dimensional contact process with infection parameter lambda we introduce a notion of sensitivity of the process to its initial condition and prove that it increases not only for values of lambda lambda c the value of the critical parameter but keeps increasing even after lambda c before finally starting to decrease for values of lambda sufficiently above lambda c this provides a counterexample to the common belief that associates maximal information transmission to criticality

the contact process on finite homogeneous trees let d be a homogeneous tree in which every vertex has d 1 neighbours where d 2 the contact process on such a tree is known to have three distinct phases we consider the process on a finite subtree namely the rooted tree of depth h and branching factor d and relate the behaviour of the process on the infinite tree to its behaviour on the finite tree for large h in the phase of strong survival we show that with probability independent of h the process on the subtree starting from a single infection survives for a time which is doubly exponential in h and almost exponential in the number of vertices of the finite tree in the phase of weak survival on the infinite tree the survival time on the finite tree is approximately linear in h in the phase of no survival the survival time on the finite tree is linear in h if one starts with all vertices initially infected and bounded by a random variable independent of h with an exponential tail if one starts from a single infection

phase transitions on nonamenable graphs we survey known results about phase transitions in various models of statistical physics when the underlying space is a nonamenable graph most attention is devoted to transitive graphs and trees

construction of the incipient infinite cluster for spread out oriented percolation above 4 1 dimensions we construct the incipient infinite cluster measure iic for sufficiently spread out oriented percolation on d for d 1 4 1 we consider two different constructions for the first construction we define n e by taking the probability of the intersection of an event e with the event that the origin is connected to x n d summing this probability over x d and normalising the sum to get a probability measure we let n and prove existence of a limiting measure the iic for the second construction we condition the connected cluster of the origin in critical oriented percolation to survive to time n and let n under the assumption that the critical survival probability is asymptotic to a multiple of n 1 we prove existence of a limiting measure with in addition we study the asymptotic behaviour of the size of the level set of the cluster of the origin and the dimension of the cluster of the origin under our methods involve minor extensions of the lace expansion methods used in a previous paper to relate critical oriented percolation to super brownian motion for d 1 4 1

stochastic domination for a hidden markov chain with applications to the contact process in a randomly evolving environment the ordinary contact process is used to model the spread of a disease in a population in this model each infected individual waits an exponentially distributed time with parameter 1 before becoming healthy in this paper we introduce and study the contact process in a randomly evolving environment here we associate to every individual an independent two state 0 1 background process given delta 0 delta 1 if the background process is in state 0 the individual if infected becomes healthy at rate delta 0 while if the background process is in state 1 it becomes healthy at rate delta 1 by stochastically comparing the contact process in a randomly evolving environment to the ordinary contact process we will investigate matters of extinction and that of weak and strong survival a key step in our analysis is to obtain stochastic domination results between certain point processes we do this by starting out in a discrete setting and then taking continuous time limits

the critical contact process seen from the right edge durrett 1984 proved the existence of an invariant measure for the critical and supercritical contact process seen from the right edge galves and presutti 1987 proved in the supercritical case that the invariant measure was unique and convergence to it held starting in any semi infinite initial state we prove the same for the critical contact process we also prove that the process starting with one particle conditioned to survive until timet converges to the unique invariant measure ast

growth processes based on 8 neighbor time delays abstract unconstrained uniform growth on a regular grid of cells starting from a single cell results in a simple centrally symmetric shape on a square grid for example the resulting clusters of cells would be diamonds rotated squares using 4 neighbor growth or squares using 8 neighbor growth in order to model growth that is directed and focused in some directions and resisted in others we formulate growth in terms of a kernel which associates a time delay with each direction on a square grid a time delay kernel tdk consists of eight elements one for each of the eight directions we give a closed form characterization of the clusters that result from tdk based growth processes specifically we show that tdks give rise to convex octagons we also show that a much richer class of cluster shapes results from the composition of tdk based growth processes where activated cells synchronously cycle through sequences of tdks lastly we provide a recovery algorithm that outputs the sequence of tdks that when composed can produce a given shape

intermediate phase for the contact process on a tree we show that in between the two critical values for the contact process on trees there are infinitely many extremal nontranslation invariant stationary distributions this conclusion is reminiscent of results of grimmett and newman for percolation on the product of a tree with the integers where there is an intermediate phase with infinitely many infinite clusters

decay of correlations in random cluster models we prove exponential decay for the tail of the radius r of the cluster at the origin for subcritical random cluster models under an assumption slightly weaker than that here d is the number of dimensions specifically if throughout the subcritical phase then for some 0 this implies the exponential decay of the two point correlation function of subcritical potts models subject to a hypothesis of at least polynomial decay of this function similar results are known already for percolation and ising models and for potts models when the number q of available states is sufficiently large indeed the hypothesis of polynomial decay has been proved rigorously for these cases in two dimensions the hypothesis that is weaker than requiring that the susceptibility be finite i e that the two point function be summable the principal new technique is a form of russo s formula for random cluster models reported by bezuidenhout grimmett and kesten for the current application this leads to an analysis of a first passage problem for random cluster models and a proof that the associated time constant is strictly positive if and only if the tail of r decays exponentially

multiplicity of phase transitions and mean field criticality on highly non amenable graphs we consider independent percolation ising and potts models and the contact process on infinite locally finite connected graphs

inhomogeneous contact processes on trees we consider an inhomogeneous contact process on a tree mathbb t k of degreek where the infection rate at any site is the death rate at any site in s subset mathbb t k is with 0 4 if s mathbb t sigma a subtree embedded in mathbb t k with 1 k then three exists c strictly between lambda c mathbb t k lambda c mathbb t sigma and 1 such that theta delta lambda c mathbb t k 0 when c and c theta delta lambda c mathbb t k 0 0 when c ifs o the origin of theta delta lambda c mathbb t k 0 then theta delta lambda c mathbb t k 0 for any e 0 1

series analysis of the generalized contact process abstract we study critical behavior in a one parameter family of generalized contact processes using series expansions our results show that the entire family belongs to a single universality class and permit a more accurate assessment of exponent uncertainties

the existence of an intermediate phase for the contact process on trees let mathbb t d be a homogeneous tree in which every vertex has d neighbors a new proof is given that the contact process on mathbb t d exhibits two phase transitions when d geq 3 a behavior which distinguishes it from the contact process on mathbb z n this is the first proof which does not involve calculation of bounds on critical values and it is much shorter than the previous proof for the binary tree mathbb t 3 the method is extended to prove the existence of an intermediate phase for a more general class of trees with exponential growth and certain symmetry properties for which no such result was previously known

critical exponents for the contact process under the triangle condition we show that a continuous time version of the triangle condition for percolation implies mean field values for several contact process critical exponents our results support the belief that the upper critical spatial dimension for the contact process is four

the complete convergence theorem of the contact process on trees consider the contact process on a homogeneous tree with degree d 3 denote by c inf p o f t i o 0 the critical value of local survival probability where o is the root of the tree pemantle and durrett and schinazi both conjectured that the complete convergence theorem should hold if c here we answer the conjecture affirmatively furthermore we will show that p o ft i o 0 at c therefore the conclusion of the complete convergence theorem cannot hold at c

sample path behaviour in connection with generalized arcsine laws letg g t t 0 be the process of last passage times at some fixed point of a markov process the dynkin lamperti theorem provides a necessary and sufficient condition forg t t to converge in law ast to some non degenerate limit which is then a generalized arcsine law under this condition we give a simple integral test that characterizes the lower functions ofg we obtain a similar result fora a t t 0 the time spent in 0 by a real valued diffusion process in connection with watanabe s recent extension of levy s second arcsine law

kinetically constrained spin models we analyze the density and size dependence of the relaxation time for kinetically constrained spin models kcsm intensively studied in the physics literature as simple models sharing some of the features of the glass transition kcsm are interacting particle systems on mathbb z d with glauber like dynamics reversible w r t a simple product i i d bernoulli p measure the essential feature of a kcsm is that the creation destruction of a particle at a given site can occur only if the current configuration around it satisfies certain constraints which completely define each specific model no other interaction is present in the model from the mathematical point of view the basic issues concerning positivity of the spectral gap inside the ergodicity region and its scaling with the particle density p remained open for most kcsm with the notably exception of the east model in d 1 aldous and diaconis in j stat phys 107 5 6 945 975 2002 here for the first time we i identify the ergodicity region by establishing a connection with an associated bootstrap percolation model ii develop a novel multi scale approach which proves positivity of the spectral gap in the whole ergodic region iii establish sometimes optimal bounds on the behavior of the spectral gap near the boundary of the ergodicity region and iv establish pure exponential decay at equilibrium for the persistence function i e the probability that the occupation variable at the origin does not change before time t our techniques are flexible enough to allow a variety of constraints and our findings disprove certain conjectures which appeared in the physical literature on the basis of numerical simulations

time dependent perturbation theory for diffusive non equilibrium lattice models the authors recently presented a perturbation theory for the asymptotic survival probability of an interacting particle system which can become trapped in an absorbing state they extend the method to a simple diffusive model analysis of the resulting series shows that diffusion is an irrelevant perturbation i e it does not change the critical behaviour quantitative predictions of the phase boundary are confirmed by results of monte carlo simulations

epidemic processes on complex networks abstract we analyse the dynamics of the sis and reed frost models on complex networks in contrast to most of the previous analytical work which relied on mean field approximations and simulations we present a probabilistic approach whereby general results are derived and illustrated on various families of graphs

the central limit theorem for supercritical oriented percolation in two dimensions we consider the cardinality of supercritical oriented bond percolation in two dimensions we show that whenever the the origin is conditioned to percolate the process appropriately normalized converges asymptotically in distribution to the standard normal law this resolves a longstanding open problem pointed out to in several instances in the literature the result applies also to the continuous time analog of the process viz the basic one dimensional contact process we also derive general random indices central limit theorems for associated random variables as byproducts of our proof

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