frozen percolation in two dimensions aldous math proc camb philos soc 128 465 477 2000 introduced a modification of the bond percolation process on the binary tree where clusters stop growing freeze as soon as they become infinite we investigate the site version of this process on the triangular lattice where clusters freeze as soon as they reach l infty diameter at least n for some parameter n we show informally speaking that in the limit n rightarrow infty the clusters only freeze in the critical window of site percolation on the triangular lattice hence the fraction of vertices that eventually i e at time 1 are in a frozen cluster tends to 0 as n goes to infinity we also show that the diameter of the open cluster at time 1 of a given vertex is with high probability smaller than n but of order n this shows that the process on the triangular lattice has a behaviour quite different from aldous process we also indicate which modifications have to be made to adapt the proofs to the case of the n parameter frozen bond percolation process on the square lattice this extends our results to the square lattice and answers the questions posed by van den berg et al random struct algorithms 40 220 226 2012

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

conformal invariance in two dimensional percolation the word percolation borrowed from the latin refers to the seeping or oozing of a liquid through a porous medium usually to be strained in this and related senses it has been in use since the seventeenth century it was introduced more recently into mathematics by s r broadbent and j m hammersley bh and is a branch of probability theory that is especially close to statistical mechanics broadbent and hammersley distinguish between two types of spreading of a fluid through a medium or between two aspects of the probabilistic models of such processes diffusion processes in which the random mechanism is ascribed to the fluid and percolation processes in which it is ascribed to the medium a percolation process typically depends on one or more probabilistic parameters for example if molecules of a gas are absorbed at the surface of a porous solid as in a gas mask then their ability to penetrate the solid depends on the sizes of the pores in it and their positions both conceived to be distributed in some random manner a simple mathematical model of such a process is often defined by taking the pores to be distributed in some regular manner that could be determined by a periodic graph and to be open thus very large or closed thus smaller than the molecules with probabilities p and 1 p as p increases the probability of deeper penetration of the gas into the interior of the solid grows

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

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

slab percolation for the ising model for the fk representation of the ising model we prove that the slab percolation threshold coincides with the critical temperature in any dimension d 3

uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation for independent translation invariant irreducible percolation models it is proved that the infinite cluster when it exists must be unique the proof is based on the convexity or almost convexity and differentiability of the mean number of clusters per site which is the percolation analogue of the free energy the analysis applies to both site and bond models in arbitrary dimension including long range bond percolation in particular uniqueness is valid at the critical point of one dimensional 1 x y 2 models in spite of the discontinuity of the percolation density there corollaries of uniqueness and its proof are continuity of the connectivity functions and except possibly at the critical point of the percolation density related to differentiability of the free energy are inequalities which bound the specific heat critical exponent in terms of the mean cluster size exponent and the critical cluster size distribution exponent e g 1 2 1 1

lax representation and quadratic first integrals for a family of non autonomous second order differential equations abstract we consider a family of non autonomous second order differential equations which generalizes the lienard equation we explicitly find the necessary and sufficient conditions for members of this family of equations to admit quadratic with the respect to the first derivative first integrals we show that these conditions are equivalent to the conditions for equations in the family under consideration to possess lax representations this provides a connection between the existence of a quadratic first integral and a lax representation for the studied dissipative differential equations which may be considered as an analogue to the theorem that connects lax integrability and arnold liouville integrability of hamiltonian systems we illustrate our results by several examples of dissipative equations including generalizations of the van der pol and duffing equations each of which have both a quadratic first integral and a lax representation

an intermediate phase with slow decay of correlations in one dimensional 1 x y 2 percolation ising and potts models we rigorously establish the existence of an intermediate ordered phase in one dimensional 1 x y 2 percolation ising and potts models the ising model truncated two point function has a power law decay exponent which ranges from its low and high temperature value of two down to zero as the inverse temperature and nearest neighbor coupling vary similar results are obtained for percolation and potts models

mean field criticality for percolation on planar non amenable graphs the critical exponents and are proved to exist and to take their mean field values for independent percolation on the following classes of infinite locally finite connected transitive graphs 1 non amenable planar with one end 2 unimodular with infinitely many ends

discontinuity of the magnetization in one dimensional 1 x y 2 ising and potts models results from percolation theory are used to study phase transitions in one dimensional ising and q state potts models with couplings of the asymptotic form j sub x y approx const absolute value x y sup 2 for translation invariant systems with well defined lim sub x infinity x sup 2 j sub x j sup possibly o or infinity we establish 1 there is no long range order at inverse temperatures beta with beta j sup less than or equal to1 2 if beta j sup q then by sufficiently increasing j sub 1 the spontaneous magnetization m is made positive 3 in models with 0 j sup infinity the magnetization is discontinuous at the transition point as originally predicted by thouless and obeys m beta sub c less than or equal to 1 beta sub c j sup sup 1 2 4 for ising q 2 models with j sup infinity it is noted that the correlation function decays as beta approx c beta absolute value x y sup 2 whenever beta beta sub c points 1 3 are deduced from previous percolation results by utilizing the fortuin kasteleyn representation which also yields other results of independent interest relating potts models with different values of q

a local limit theorem for triple connections in subcritical bernoulli percolation we prove a local limit theorem for the probability of a site to be connected by disjoint paths to three points in subcritical bernoulli percolation on mathbb z d d geq2 in the limit where their distances tend to infinity

on the critical percolation probabilities we prove that the critical probabilities of site percolation on the square lattice satisfy the relation p c p 1 furthermore we prove the continuity of the function percolation probability

non perturbative criteria for gibbsian uniqueness for spin systems with an internal symmetry we provide sufficient conditions for unicity of the gibbs state and or complete analyticity by comparison to random cluster models

subcritical percolation with a line of defects we consider the bernoulli bond percolation process pp p on the nearest neighbor edges of zd which are open independently with probability p pc except for those lying on the first coordinate axis for which this probability is p define p p limn n 1logpp p 0 ne1 and p p p we show that there exists p c p c p d such that p p p if p p c moreover p c p 2 p c p 3 p and p c p d p for d 4 we also analyze the behavior of p p p as p p c in dimensions d 2 3 finally we prove that when p p c the following purely exponential asymptotics holds pp p 0 ne1 de p p n 1 o 1 for some constant d d p p uniformly for large values of n this work gives the first results on the rigorous analysis of pinning type problems that go beyond the effective models and don t rely on exact computations

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

a sharp transition for the two dimensional ising percolation we show that the percolation transition for the two dimensional ising model is sharp namely we show that for every reciprocal temperature 0 there exists a critical valueh c of external magnetic fieldh such that the following two statements hold we also shows that the percolation probability is continuous in h except on the half line 0 c

invasion percolation and the incipient infinite cluster in 2d we establish two links between two dimensional invasion percolation and kesten s incipient infinite cluster iic we first prove that the kth moment of the number of invaded sites within the box n n n n is of order n2 n k for k 1 where n is the probability that the origin in critical percolation is connected to the boundary of a box of radius n this improves a result of y zhang we show that the size of the invaded region when scaled by n2 n is tight

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

a simple proof of exponential decay of subcritical contact processes this paper gives a new simple proof of the known fact that for contact processes on general lattices in the subcritical regime the expected number of infected sites decays exponentially fast as time tends to infinity the proof also yields an explicit bound on the survival probability below the critical recovery rate which shows that the critical exponent associated with this function is bounded from below by its mean field value the main idea of the proof is that if the expected number of infected sites decays slower than exponentially then this implies the existence of a harmonic function that can be used to show that the process survives for any lower value of the recovery rate

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 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

surface order large deviations for high density percolation we derive surface order large deviation estimates for the volume of the largest cluster and for the volume of the largest region surrounded by a cluster of a bernoulli percolation process restricted to a big finite box with sufficiently large parameter we also establish a useful version of the isoperimetric inequality which is the main tool of our proofs

on the range of bond percolation thresholds for fully triangulated graphs this paper constructs a sequence of planar periodic fully triangulated lattices which have bond percolation thresholds tending to zero as a consequence we see that bond percolation thresholds of such graphs can range from 0 to 2 sin 18 0 3473 which is the threshold of the triangular lattice rigorous bounds and simulations for other fully triangulated lattices suggest that the threshold for the triangular lattice is the largest for this class of graphs

discontinuity of the percolation density in one dimensional 1 x y 2 percolation models we consider one dimensional percolation models for which the occupation probability of a bond kx y has a slow power decay as a function of the bond s length for independent models and with suitable reformulations also for more general classes of models it is shown that i no percolation is possible if for short bondskx y p 1 this dichotomy resembles one for the magnetization in 1 x y 2 ising models which was first proposed by thouless and further supported by the renormalization group flow equations of anderson yuval and hamann the proofs of the above percolation phenomena involve rigorous renormalization type arguments of a different sort

the q state potts model in the standard pirogov sinai theory surface tensions and wilson loops theq state potts model both scalar and gauge versions is rewritten with the help of the duality transformation into a form of the pirogov sinai theory with noninteracting contours that can be controlled by cluster expansions onceq is large enough this is then used in a new proof of the existence of a unique transition inverse temperature t where the mean internal energy is discontinuous moreover we prove for the scalar model again forq large enough that there are discontinuities at t of the magnetization and of the mass gap with the magnetization vanishing below t and the mass gap vanishing above t we also show that the surface tensions between ordered stable phases are strictly positive up to t and the surface tension between an ordered phase and the disordered one is strictly positive at t for the three dimensional gauge model the wilson parameter exhibits a direct transition from an area law decay quark confinement to a perimeter law decay deconfinement

phase diagram of inhomogeneous percolation with a defect plane let mathbb l be the d dimensional hypercubic lattice and let mathbb l 0 be an s dimensional sublattice with 2 le s p c d and a surface supercritical regime for p sigma p we show that sigma p is a strictly decreasing function for p in 0 p c d with a jump discontinuity at p c d we extend the aizenman barsky differential inequalities for homogeneous percolation to the inhomogeneous model and use them to prove that the susceptibility is finite inside the subcritical phase we prove that the cluster size distribution decays exponentially in the subcritical phase and sub exponentially in the supercritical phases for a model of lattice animals with a defect plane the free energy is related to functions of the inhomogeneous percolation model and we show how the percolation transition implies a non analyticity in the free energy of the animal model finally we present simulation estimates of the critical curve sigma p

critical percolation of virtually free groups and other tree like graphs this article presents a method for finding the critical probability p c for the bernoulli bond percolation on graphs with the so called tree like structure such a graph can be decomposed into a tree of pieces each of which has finitely many isomorphism classes this class of graphs includes the cayley graphs of amalgamated products hnn extensions or general groups acting on trees it also includes all transitive graphs with more than one end the idea of the method is to find a multi type galton watson branching process with a parameter p which has finite expected population size if and only if the expected percolation cluster size is finite this provides sufficient information about p c in particular if the pairwise intersections of pieces are finite then p c is the smallest positive p such that det m 1 0 where m is the first moment matrix of the branching process if the pieces of the tree like structure are finite then p c is an algebraic number and we give an algorithm computing p c as a root of some algebraic function we show that any cayley graph of a virtually free group i e a group acting on a tree with finite vertex stabilizers with respect to any finite generating set has a tree like structure with finite pieces in particular we show how to compute p c for the cayley graph of a free group with respect to any finite generating set

the phase transitions of the planar random cluster and potts models with q ge 1 are sharp we prove that random cluster models with q ge 1 on a variety of planar lattices have a sharp phase transition that is that there exists some parameter p c below which the model exhibits exponential decay and above which there exists a s an infinite cluster the result may be extended to the potts model via the edwards sokal coupling our method is based on sharp threshold techniques and certain symmetries of the lattice in particular it makes no use of self duality part of the argument is not restricted to planar models and may be of some interest for the understanding of random cluster and potts models in higher dimensions due to its nature this strategy could be useful in studying other planar models satisfying the fkg lattice condition and some additional differential inequalities

one dimensional 1 j i s percolation models the existence of a transition for s 2 consider a one dimensional independent bond percolation model withpj denoting the probability of an occupied bond between integer sitesi andi j j 1 ifpj is fixed forj 2 and mathop lim limits j to infty j2pj 1 then unoriented percolation occurs forp1 sufficiently close to 1 this result analogous to the existence of spontaneous magnetization in long range one dimensional ising models is proved by an inductive series of bounds based on a renormalization group approach using blocks of variable size oriented percolation is shown to occur forp1 close to 1 if mathop lim limits j to infty jspj 0 for somes 2 analogous results are valid for one dimensional site bond percolation models

the supercritical phase of percolation is well behaved we prove a general result concerning the critical probabilities of subsets of a lattice l it is a consequence of this result that the critical probability of a percolation process on l equals the limit of the critical probability of a slice of l as the thickness of the slice tends to infinity this verification of one of the standard hypotheses of the subject settles many questions concerning supercritical percolation

equality of directional critical exponents in multiparameter percolation models in multiparameter percolation models a directional critical exponent may be considered for each direction of approach in the parameter space to a point on the critical surface an elementary coupling argument shows that for each point on the critical surface the directional critical exponents are equal for all directions strictly in the positive orthant using the substitution method the set of directions with equal directional critical exponents can he extended beyond the orthant the anisotropic triangular lattice bond percolation model is used to illustrate the method

on the chemical distance for supercritical bernoulli percolation we prove large deviation estimates at the correct order for the graph distance of two sites lying in the same cluster of an independent percolation process we improve earlier results of gartner and molchanov and grimmett and marstrand and answer affirmatively a conjecture of kozlov

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

decay properties of the connectivity for mixed long range percolation models on z d in this short note we consider mixed short long range independent bond percolation models on k d let puv be the probability that the edge u v will be open allowing a x y dependent length scale and using a multi scale analysis due to aizenman and newman we show that the long distance behavior of the connectivity xy is governed by the probability pxy the result holds up to the critical point

critical points and intermediate phases on wedges of zd the authors examine the phase structure of self avoiding walks the ising magnet and bond percolation defined on subsets of zd d or 2 which have the geometry of wedges they prove that if the cross sectional area of the wedge diverges with its width then the high temperature critical point of any of these models defined on the wedge coincides with that of the corresponding model on the full lattice for the ising magnet and bond percolation they show that there is a non trivial low temperature critical point if the cross sectional area of the wedge diverges logarithmically with its width moreover for bond percolation they show that the low temperature critical point of a logarithmic wedge may be made arbitrarily close to that of the full lattice by taking the coefficient of the logarithm large enough corollaries to these theorems include the existence of an intermediate phase for the ising magnet and bond percolation on logarithmic wedges and the existence of a first order transition for percolation models on a subclass of these wedges

critical percolation on a bethe lattice revisited we introduce the model of independent percolation on general graphs with emphasis on the bethe lattice for which we prove the existence of a phase transition with the precise calculation of the critical value and derive the value of the critical exponents gamma and b

bernoulli percolation above threshold an invasion percolation analysis on utilise le processus de percolation d invasion pour demontrer des resultats relatifs a la percolation de bernoulli sur z d

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

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

phase separation in random cluster models ii the droplet at equilibrium and local deviation lower bounds we study the droplet that results from conditioning the planar subcritical fortuin kasteleyn random cluster model on the presence of an open circuit 0 encircling the origin and enclosing an area of at least or exactly n2 we consider local deviation of the droplet boundary measured in a radial sense by the maximum local roughness mlr 0 this being the maximum distance from a point in the circuit 0 to the boundary conv 0 of the circuit s convex hull and in a longitudinal sense by what we term maximum facet length namely the length of the longest line segment of which the polygon conv 0 is formed we prove that there exists a constant c 0 such that the conditional probability that the normalised quantity n 1 3 log n 2 3 mlr 0 exceeds c tends to 1 in the high n limit and that the same statement holds for n 2 3 log n 1 3 mfl 0 to obtain these bounds we exhibit the random cluster measure conditional on the presence of an open circuit trapping high area as the invariant measure of a markov chain that resamples sections of the circuit boundary we analyse the chain at equilibrium to prove the local roughness lower bounds alongside complementary upper bounds provided in 14 the fluctuations mlr 0 and mfl 0 are determined up to a constant factor

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

phase separation in random cluster models iii circuit regularity we study the droplet that results from conditioning the subcritical fortuin kasteleyn random cluster model on the presence of an open circuit encircling the origin and enclosing an area of at least or exactly n 2 in this paper we prove that the resulting circuit is highly regular we define a notion of a regeneration site in such a way that for any such site v the circuit cuts through the radial line segment through v only at v we show that provided that the conditioned circuit is centred at the origin in a natural sense the set of regeneration sites reaches into all parts of the circuit with maximal distance from one such site to the next being at most logarithmic in n with high probability the result provides a flexible control on the conditioned circuit that permits the use of surgical techniques to bound its fluctuations and as such it plays a crucial role in the derivation of bounds on the local fluctuation of the circuit carried out in hammond arxiv 1001 1527 2009 arxiv 1001 1528 2009

gaussian fluctuations of connectivities in the subcritical regime of percolation we consider thed dimensional bernoulli bond percolation model and prove the following results for allp pc the finite volume conditional measures defined by requiring the existence of a dual path between opposing faces of the boundary converge in the infinite volume limit to the standard bernoulli measure

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

carrousel in family and break non isolated hypersurface singularities in ℂ3 we study the boundary l t of the milnor fiber for the reduced holomorphic germs f bbb c 3 0 rightarrow bbb c 0 having a non isolated singularity at 0 we prove that l t is a graph manifold by using a new technique of carrousels depending on one parameter our results enable us to compare the topology of l t and of the link of the normalization of f 1 0

approximate zero one laws and sharpness of the percolation transition in a class of models including two dimensional ising percolation one of the most well known classical results for site percolation on the square lattice is the equation p c p c 1 in words this equation means that for all values p c of the parameter p the following holds either a s there is an infinite open cluster or a s there is an infinite closed star cluster this result is closely related to the percolation transition being sharp below p c the size of the open cluster of a given vertex is not only a s finite but has a distribution with an exponential tail the analog of this result has been proven by higuchi in 1993 for two dimensional ising percolation at fixed inverse temperature c with external field h the parameter of the model using sharp threshold results approximate zero one laws and a modification of an rsw like result by bollobas and riordan we show that these results hold for a large class of percolation models where the vertex values can be nicely represented in a sense which will be defined precisely by i i d random variables we point out that the ordinary percolation model obviously belongs to this class and we also show that the ising model mentioned above belongs to it

percolation thresholds on two dimensional voronoi networks and delaunay triangulations the site percolation threshold for the random voronoi network is determined numerically with the result pc 0 714 10 0 000 02 using monte carlo simulation on periodic systems of up to 40 000 sites the result is very close to the recent theoretical estimate pc approximately 0 7151 of neher for the bond threshold on the voronoi network we find pc 0 666 931 0 000 005 implying that for its dual the delaunay triangulation pc 0 333 069 0 000 005 these results rule out the conjecture by hsu and huang that the bond thresholds are 2 3 and 1 3 respectively but support the conjecture of wierman that for fully triangulated lattices other than the regular triangular lattice the bond threshold is less than 2 sin pi 18 approximately 0 3473

the random walk representation of classical spin systems and correlation inequalities ferromagnetic lattice spin systems can be expressed as gases of random walks interacting via a soft core repulsion by using a mixed spinrandom walk representation we present a unified approach to many recently established correlation inequalities as an application of these inequalities we obtain a simple proof of the mass gap for the 4 2 quantum field model we also establish new upper bounds on critical temperatures

divergence of shape fluctuations in two dimensions we consider stochastic growth models such as standard first passage percolation on z d where to leading order there is a linearly growing deterministic shape under natural hypotheses we prove that for d 2 the shape fluctuations grow at least logarithmically in all directions although this bound is far from the expected power law behavior with exponent 1 3 it does prove divergence with additional hypotheses we obtain inequalities involving and the related exponent which is expected to equal 2 3 for d 2 combining these inequalities with previously known results we obtain for standard first passage percolation the bounds 1 8 for d 2 and 3 4 for all d

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

coexistence of partially disordered ordered phases in an extended potts model we consider a generalization of the standard potts model in which there are q r s states with an interaction that distinguishes the two subspecies we develop a graphical representation of the fk type for the system and show that this representation may be incorporated directly into reflection positivity arguments using combinations of these techniques we establish detailed properties of the phase diagram including the existence of sharp triple points whenever relevant the phases are characterized by the percolation properties of the underlying representation

phase separation in random cluster models i uniform upper bounds on local deviation this is the first in a series of three papers that addresses the behaviour of the droplet that results in the percolating phase from conditioning the planar fortuin kasteleyn random cluster model on the presence of an open dual circuit 0 encircling the origin and enclosing an area of at least or exactly n2 by the fortuin kasteleyn representation the model is a close relative of the droplet formed by conditioning the potts model on an excess of spins of a given type we consider local deviation of the droplet boundary measured in a radial sense by the maximum local roughness mlr 0 this being the maximum distance from a point in the circuit 0 to the boundary of the circuit s convex hull and in a longitudinal sense by what we term maximum facet length mlf 0 namely the length of the longest line segment of which the polygon is formed the principal conclusion of the series of papers is the following uniform control on local deviation that there are constants 0 c c such that the conditional probability that the normalized quantity n 1 3 log n 2 3mlr lies in the interval c c tends to 1 in the high n limit and that the same statement holds for n 2 3 log n 1 3 mlf in this way we confirm the anticipated n1 3 scaling of maximum local roughness and provide a sharp logarithmic power law correction this local deviation behaviour occurs by means of locally gaussian effects constrained globally by curvature and we believe that it arises in many radially defined stochastic interface models including growth models belonging to the kardar parisi zhang universality class

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

decay of correlations in nearest neighbor self avoiding walk percolation lattice trees and animals we consider nearest neighbor self avoiding walk bond percolation lattice trees and bond lattice animals on z d the two point functions of these models are respectively the generating function for self avoiding walks from the origin to x z d the probability of a connection from the origin to x and the generating functions for lattice trees or lattice animals containing the origin and x using the lace expansion we prove that the two point function at the critical point is asymptotic to const x 2 d as x for d 5 for self avoiding walk for d 19 for percolation and for sufficiently large d for lattice trees and animals these results are complementary to those of ann probab 31 2003 349 408 where spread out models were considered in the course of the proof we also provide a sufficient and rather sharp if d 4 condition under which the two point function of a random walk on z d is asymptotic to const x 2 d as x

critical two point functions and the lace expansion for spread out high dimensional percolation and related models we consider spread out models of self avoiding walk bond percolation lattice trees and bond lattice animals on mathbb z d having long finite range connections above their upper critical dimensions d 4 self avoiding walk d 6 percolation and d 8 trees and animals the two point functions for these models are respectively the generating function for self avoiding walks from the origin to x in mathbb z d the probability of a connection from 0 to x and the generating function for lattice trees or lattice animals containing 0 and x we use the lace expansion to prove that for sufficiently spread out models above the upper critical dimension the two point function of each model decays at the critical point as a multiple of x 2 d as x to infty we use a new unified method to prove convergence of the lace expansion the method is based on x space methods rather than the fourier transform our results also yield unified and simplified proofs of the bubble condition for self avoiding walk the triangle condition for percolation and the square condition for lattice trees and lattice animals for sufficiently spread out models above the upper critical dimension

the triangle condition for percolation aizenman and newman introduced an unverified condition the triangle condition which has been shown to imply that a number of percolation critical exponents take their mean field values and which is expected to hold above six dimensions for nearest neighbour percolation we prove that the triangle condition is satisfied in sufficiently high dimensions for the nearest neighbour model and above six dimensions for a class of spread out models the proof uses an expansion which is related to the lace expansion for self avoiding walk percolation is a simple probabilistic model for which many interesting problems remain unsolved the basic objects in percolation are random graphs on an infinite lattice typically there is a critical density for the graph edges known as bonds below which there is zero probability that a fixed point in the lattice is part of an infinite connected graph but above which this probability is strictly positive this abrupt change in behaviour has been used in chemistry and statistical physics to model phase transitions in a variety of physical systems such as fluid flow through a porous medium hence the name percolation random resistor networks and the gelation of branched polymers for an introduction see 15 20 to define the models we are interested in we consider the of dimensional integer lattice z whose elements are referred to as sites and the set of pairs b x y of distinct sites the bonds to each bond b is associated an independent bernoulli random variable nb9 with nb 1 with probability p jb and nb 0 with probability p jh the jb are fixed and z invariant and p is a parameter the nearest neighbour model is defined by taking jb 1 for those b x y such that x y i 1 and jb 0 otherwise expectation with respect to the joint distribution of the nb is denoted by if nb 1 we say b is occupied and otherwise b is vacant given a bond configuration nb two sites x and y are said to be connected if there is a path from x to y which consists of occupied bonds the set of sites which are connected to x is called the connected cluster of x and is denoted c x the probability that x is connected to y is written tp x y 1 ip x y i x and y are connected i y e c x p where i denotes the indicator function denoting by c x the number of sites in c x the expected cluster size or susceptibility is given by

exponential decay of connection probabilities for subcritical voronoi percolation in mathbb r d r d we prove that for voronoi percolation on mathbb r d d ge 2 there exists p c p c d in 0 1 such that for dimension 2 this result offers a new way of showing that p c 2 1 2 this paper belongs to a series of papers using the theory of algorithms to prove sharpness of the phase transition see 10 11

convergence to equilibrium of random ising models in the griffiths phase we consider glauber type dynamics for disordered ising spin systems with nearest neighbor pair interactions in the griffiths phase we prove that in a nontrivial portion of the griffiths phase the system has exponentially decaying correlations of distant functions with probability exponentially close to 1 this condition has in turn been shown elsewhere to imply that the convergence to equilibrium is faster than any stretched exponential and that the average over the disorder of the time autocorrelation function goes to equilibrium faster than exp k log t d d 1 we then show that for the diluted ising model these upper bounds are optimal

on the critical behavior of the magnetization in high dimensional ising models we derive rigorously general results on the critical behavior of the magnetization in ising models as a function of the temperature and the external field for the nearest neighbor models it is shown that ind 4 dimensions the magnetization is continuous att c and its critical exponents take the classical values 3 and 1 2 with possible logarithmic corrections atd 4 the continuity and other explicit bounds formally extend tod 3 1 2 other systems to which the results apply include long range models ind 1 dimension with 1 x y couplings for which 2 1 replacesd in the above summary the results are obtained by means of differential inequalities derived here using the random current representation which is discussed in detail for the case of a nonvanishing magnetic field

convergence of percolation probability functions to cumulative distribution functions on square lattices with 1 0 neighborhood abstract we consider a percolation model on square lattices with sites weighted by beta distributed random variables s beta a b with a positive real parameters a 0 and b 0 using the monte carlo method we estimate the percolation probability p as a relative frequency p averaged over the target subset of sites on a square lattice as a result of the comparative analysis we formulate two empirical hypotheses the first on the correspondence of percolation thresholds p c to p 0 quantiles where level p 0 0 592746 coincides with the percolation threshold for the site percolation model on a square lattice of random variables s i weighing sites of the square lattice and the second on the convergence of statistical estimates of percolation probability functions p p to cumulative distribution functions f s i p of these variables s i for the supercritical values of the occupation probability p p c

quenched invariance principles for random walks and elliptic diffusions in random media with boundary via a dirichlet form extension theorem and making full use of two sided heat kernel estimates we establish quenched invariance principles for random walks in random environments with a boundary in particular we prove that the random walk on a supercritical percolation cluster or amongst random conductances bounded uniformly from below in a half space quarter space etc converges when rescaled diffusively to a reflecting brownian motion which has been one of the important open problems in this area we establish a similar result for the random conductance model in a box which allows us to improve existing asymptotic estimates for the relevant mixing time furthermore in the uniformly elliptic case we present quenched invariance principles for domains with more general boundaries

sharpness of the phase transition in percolation models the equality of two critical points the percolation threshold ph and the point p where the cluster size distribution ceases to decay exponentially is proven for all translation invariant independent percolation models on homogeneous d dimensional lattices d 1 the analysis is based on a pair of new nonlinear partial differential inequalities for an order parameter m h which for h q reduces to the percolation density p at the bond density p l e in the single parameter case these are 1 m hdm dh m2 mdm d and 2 dm d j mdm dh inequality 1 is intriguing in that its derivation provides yet another hint of a 3 structure in percolation models moreover through the elimination of one of its derivatives 1 yields a pair of ordinary differential inequalities which provide information on the critical exponents and one of these resembles an ising model inequality of fr hlich and sokal and yields the mean field bound 5 2 and the other implies the result of chayes and chayes that an inequality identical to 2 is known for ising models where it provides the basis for newman s universal relation 5 1 1 and for certain extrapolation principles which are now made applicable also to independent percolation these results apply to both finite and long range models with or without orientation and extend to periodic and weakly inhomogeneous systems

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