system analysis via integral quadratic constraints this paper introduces a unified approach to robustness analysis with respect to nonlinearities time variations and uncertain parameters from an original idea by yakubovich 1967 the approach has been developed under a combination of influences from the western and russian traditions of control theory it is shown how a complex system can be described using integral quadratic constraints iqc for its elementary components a stability theorem for systems described by iqcs is presented that covers classical passivity dissipativity arguments but simplifies the use of multipliers and the treatment of causality a systematic computational approach is described and relations to other methods of stability analysis are discussed last but not least the paper contains a summarizing list of iqcs for important types of system components

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

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

a structure theorem for boolean functions with small total influences we show that on every product probability space boolean functions with small total inuences are essentially the ones that are almost measurable with respect to certain natural sub sigma algebras this theorem in particular describes the structure of monotone set properties that do not exhibit sharp thresholds our result generalizes the core of friedgut s seminal work on properties of random graphs to the setting of arbitrary boolean functions on general product probability spaces and improves the result of bourgain in his appendix to friedgut s paper

an upper bound on the critical percolation probability for the three dimensional cubic lattice we prove that the critical probability for site percolation on the three dimensional cubic lattice satisfies the inequality p 3 c 1 2 an application to the three dimensional ising model is given

strict inequalities for some critical exponents in two dimensional percolation for 2d percolation we slightly improve a result of chayes and chayes to the effect that the critical exponent for the percolation probability isstrictly less than 1 the same argument is applied to prove that ifl x y x r cos y r sin for some r 0 or and limp p c log p p c 1 log pcr ito is connected to by an occupied path inl then is strictly decreasing in on 0 2 similarly limn logn 1 logp cr ito is connected by an occupied path inl to the exterior of n n n n is strictly decreasing in on 0 2

crabgrass measles and gypsy moths an introduction to modern probability on etudie cinq modeles qui permettent d expliquer des processus lies aux systemes de particules en interaction ainsi que d autres processus

on system identification of complex systems from finite data we introduce a new principle for identification based on choosing a model from the model parameterization which best approximates the unknown real system belonging to a more complex space of systems that do not lend themselves to a finite parameterization the principle is particularly effective for robust control as it leads to a precise notion of parametric and nonparametric error and the identification problem can be equivalently perceived as that of robust convergence of the parameters in the face of unmodeled errors the main difficulty in its application stems from the interplay of noise and unmodeled dynamics and requires developing novel two step algorithms that amount to annihilation of the unmodeled error followed by averaging out the noise the paper establishes robust convergence for a large class of systems topologies and unmodeled errors sample path based finite time polynomial rate of convergence and annihilation correlation algorithms for linearly parameterized model structures

renormalisation group approach to delaunay percolation networks with topological disorder a renormalisation group approach is developed for delaunay percolating systems in two and three dimensions using a scaling transformation for a finite lattice in real space considering various renormalisation transformations for two and three dimensional delaunay lattices the authors determine the behaviour of the probabilities under a scale transformation and calculate the fixed point and connectedness length exponent the fixed points for the two dimensional bond lattice and the three dimensional site lattice are 0 3229 and 0 1443 respectively which are in excellent agreement with results of monte carlo simulations the fixed point for the two dimensional site lattice gives the value 1 2 for the critical percolation probability which is equal to the known result for a fully triangulated lattice

a lower bound for the critical probability of the square lattice site percolation we prove by elementary combinatorial considerations that the critical probability of the square lattice site percolation is larger than 0 503478

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

the large n limit of the threshold values in mandelbrot s fractal percolation process the authors consider mandelbrot s fractal percolation process characterised by a density parameter p in 0 1 and an integer subdivision index n 1 for each n the process is known to have a percolation transition at density pc n in 0 1 they prove that lim pc n pc where pc is the threshold value of the ordinary square lattice site percolation model

broadcasting on random directed acyclic graphs we study the following generalization of the well known model of broadcasting on trees consider an infinite directed acyclic graph dag with a unique source vertex x let the collection of vertices at distance k from x be called the k th layer and suppose every non source vertex has indegree d geq 2 at layer 0 the source vertex is given a random bit at layer k geq 1 each vertex receives d bits from its parents in the k 1 th layer which are transmitted along edges that are independent binary symmetric channels bscs with crossover probability delta in left 0 frac 1 2 right each vertex combines its d noisy inputs using a deterministic d ary boolean processing function that generates the value at the vertex the goal is to be able to reconstruct the original bit x with probability of error bounded away from frac 1 2 using the values of all vertices at an arbitrarily deep layer k this question is closely related to models of reliable computation and storage and information flow in biological networks in this paper we treat the case of randomly constructed dags for which we show that broadcasting is only possible if the bsc noise level delta is below a certain degree and function dependent critical threshold for d geq 3 and random dags with layers of size omega log k and majority processing functions we identify the critical threshold for d 2 we establish a similar result for the nand processing function we also prove a partial converse result for odd d geq 3 illustrating that the identified thresholds are impossible to improve by selecting different processing functions if the decoder is restricted to using a single vertex s value finally for any bsc noise level delta we construct explicit dags using regular bipartite lossless expander graphs with bounded degree and layers of size theta log k admitting reconstruction in particular we show that the first r layers of such dags can be generated in either deterministic quasi polynomial time or randomized polylogarithmic time in r these results portray a doubly exponential advantage for storing a bit in bounded degree dags compared to trees where d 1 but layer sizes need to grow exponentially with depth in order for broadcasting to be possible

strict inequalities of critical values in continuum percolation we consider the supercritical finite range random connection model where the points x y of a homogeneous planar poisson process are connected with probability f y x for a given f performing percolation on the resulting graph we show that the critical probabilities for site and bond percolation satisfy the strict inequality p c mathrm site p c mathrm bond we also show that reducing the connection function f strictly increases the critical poisson intensity

quenched survival of bernoulli percolation on galton watson trees we explore the survival function for percolation on galton watson trees letting g t p represent the probability a tree t survives bernoulli percolation with parameter p we establish several results about the behavior of the random function g mathbf t cdot where mathbf t is drawn from the galton watson distribution these include almost sure smoothness in the supercritical region an expression for the k text th order taylor expansion of g mathbf t cdot at criticality in terms of limits of martingales defined from mathbf t this requires a moment condition depending on k and a proof that the k text th order derivative extends continuously to the critical value each of these results is shown to hold for almost every galton watson tree

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

infinite ab percolation clusters exist on the triangular lattice infinite ab percolation clusters are shown to exist almost surely on the triangular lattice for an interval of parameter values it was not previously known that infinite ab percolation occurred with positive probability on any planar lattice

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

subsequential scaling limits for liouville graph distance for 0 gamma 2 and delta 0 we consider the liouville graph distance which is the minimal number of euclidean balls of gamma liouville quantum gravity measure at most delta whose union contains a continuous path between two endpoints in this paper we show that the renormalized distance is tight and thus has subsequential scaling limits at delta rightarrow 0 in particular we show that for all delta 0 the diameter with respect to the liouville graph distance has the same order as the typical distance between two endpoints

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

connectivity properties of mandelbrot s percolation process in 1974 mandelbrot introduced a process in 0 1 2 which he called canonical curdling and later used in this book s on fractals to generate self similar random sets with hausdorff dimension d 0 2 in this paper we will study the connectivity or percolation properties of these sets proving all of the claims he made in sect 23 of the fractal geometry of nature and a new one that he did not anticipate there is a probability pc 0 1 so that if p pc then the set is duslike i e the largest connected component is a point whereas if p pc notice the opposing sides are connected with positive probability and furthermore if we tile the plane with independent copies of the system then there is with probability one a unique unbounded connected component which intersects a positive fraction of the tiles more succinctly put the system has a first order phase transition

infinite ab percolation clusters exist the authors provide an example of a lattice on which infinite ab percolation clusters can occur answering a question of halley they identify parameter values for which infinite ab percolation occurs and does not occur on the lattice

on a sharp transition from area law to perimeter law in a system of random surfaces we introduce and study a phase transition which is associated with the spontaneous formation of infinite surface sheets in a bernoulli system of random plaquettes the transition is manifested by a change in the asymptotic behavior of the probability of the formation of a surface spanning a prescribed loop as such this transition offers a generalization of the bond percolation phenomenon at low plaquette densities the probability for large loops is shown to decay exponentially with the loops area whereas for high densities the decay is by a perimeter law furthermore we show that the two phases of the three dimensional plaquette system are in a precise correspondence with the two phases of the dual system of random bonds thus if a natural conjecture about the phase structure of the bond percolation model is true then there is a sharp transition in the asymptotic behavior of the surface events our analysis incorporates block variables in terms of which a non critical system is transformed into one which is close to a trivial high or low density fixed point stochastic geometric effects like those discussed here play an important role in lattice gauge theories

large deviations for the contact process and two dimensional percolation summary the following results are proved 1 for the upper invariant measure of the basic one dimensional supercritical contact process the density of l s has the usual large deviation behavior the probability of a large deviation decays exponentially with the number of sites considered 2 for supercritical two dimensional nearest neighbor site or bond percolation the density ya of sites inside a square a which belong to the infinite cluster has the following large deviation properties the probability that ya deviates from its expected value by a positive amount decays exponentially with the area of a while the probability that it deviates from its expected value by a negative amount decays exponentially with the perimeter of a these two problems are treated together in this paper because similar techniques renormalization are used for both

upper bounds on fourier entropy given a function f 0 1 n rightarrow mathbb r its fourier entropy is defined to be sum s widehat f 2 s log widehat f 2 s where hat f denotes the fourier transform of f this quantity arises in a number of applications especially in the study of boolean functions an outstanding open question is a conjecture of friedgut and kalai 1996 called the fourier entropy influence fei conjecture asserting that the fourier entropy of any boolean function f is bounded above up to a constant factor by the total influence average sensitivity of f

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

influence and sharp threshold theorems for monotonic measures the influence theorem for product measures on the discrete space 0 1 n may be extended to probability measures with the property of monotonicity which is equivalent to strong positive association corresponding results are valid for probability measures on the cube 0 1 n that are absolutely continuous with respect to lebesgue measure these results lead to a sharp threshold theorem for measures of random cluster type and this may be applied to box crossings in the two dimensional random cluster model

discontinuity of the wilson string tension in the 4 dimensional lattice pure gauge potts model we consider the 4 dimensionalq state pure gauge potts model forq large enough we give a new proof of the existence of a unique coupling constant t where a first order phase transition occurs moreover we prove the following new results the string tension is discontinuous at t the wilson parameter exhibits at t a direct transition from an area law decay quark confinement to a perimeter law decay quark deconfinement

crossing probabilities for voronoi percolation we prove that the standard russo seymour welsh theory is valid for voronoi percolation this implies that at criticality the crossing probabilities for rectangles are bounded by constants depending only on their aspect ratio this result has many consequences such as the polynomial decay of the one arm event at criticality

convolution quadrature method based symmetric galerkin boundary element method for 3 d elastodynamics the boundary integral equations in 3 d elastodynamics contain convolution integrals with respect to the time they can be performed analytically or with the convolution quadrature method the latter time stepping procedure s benefit is the usage of the laplace transformed fundamental solution therefore it is possible to apply this method also to problems where analytical time dependent fundamental solutions might not be known to obtain a symmetric formulation the second boundary integral equation has to be used which unfortunately requires special care in the numerical implementation since it involves hypersingular kernel functions therefore a regularization for closed surfaces of the laplace transformed elastodynamic kernel functions is presented which transforms the bilinear form of the hypersingular integral operator to a weakly singular one supplementarily a weakly singular formulation of the laplace transformed elastodynamic double layer potential is presented this results in a time domain boundary element formulation involving at least only weakly singular integral kernels finally numerical studies validate this approach with respect to different spatial and time discretizations further a comparison with the wider used collocation method is presented it is shown numerically that the presented formulation exhibits a good convergence rate and a more stable behavior compared with collocation methods copyright 2008 john wiley sons ltd

on robust control synthesis and analysis in a hilbert space the motivation for this paper stems from the need to develop a uniform framework for addressing problems in identi cation and robust control system identi cation for in nite dimensional hilbert spaces has been addressed earlier by the authors system identi cation set in an hilbert space results in uncertain models where the description of non parametric error is typically a ball belonging to the hilbert space the scope of this paper is to complement these results develop robust control synthesis and analysis results for some special yet important cases in this paper we derive a convex parameterization of robustly stabilizing controllers for lti discrete time systems de ned on hilbert spaces the perturbations are of rank one type having both real parametric and non parametric components the parameterization allows for imposing other constraints to obtain meaningful performance from the controller analysis tools are also developed for robust stability under siso block diagonally structured perturbations the robustness analysis problem reduces to a nite dimensional lmi veri cation which makes the procedure extremely e cient c

sharp thresholds of graph properties and the sat problem consider g n p to be the probability space of random graphs on n vertices with edge probability p we will be considering subsets of this space defined by monotone graph properties a monotone graph property p is a property of graphs such that a p is invariant under graph automorphisims b if graph h has property p then so does any graph g having h as a subgraph a monotone symmetric family of graphs is a family defined by such a property one of the first observations made about random graphs by erdos and renyi in their seminal work on random graph theory 12 was the existence of threshold phenomena the fact that for many interesting properties p the probability of p appearing in g n p exhibits a sharp increase at a certain critical value of the parameter p bollobas and thomason proved the existence of threshold functions for all monotone set properties 6 and in 14 it is shown that this behavior is quite general and that all monotone graph properties exhibit threshold behavior i e the probability of their appearance increases from values very close to 0 to values close to 1 in a very small interval more precise analysis of the size of the threshold interval is done in 7 this threshold behavior which occurs in various settings which arise in combinatorics and computer science is an instance of the phenomenon of phase transitions which is the subject of much interest in statistical physics one of the main questions that arises in studying phase transitions is how sharp is the transition for example one of the motivations for this paper arose from the question of the sharpness of the phase transition for the property of satisfiability of a random kcnf boolean formula nati linial who introduced me to this problem suggested that although much concrete analysis was being performed on this problem the best approach would be to find general conditions for sharpness of the phase transition answering the question posed in 14 as to the relation between the length of the threshold interval and the value of the critical probability in this paper we indeed introduce a simple condition and prove it is sufficient stated roughly in the setting of random graphs the main theorem states that if a property has a coarse threshold then it can be approximated by the property of having certain given graphs as a subgraph this condition can be applied in a more

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

another critical exponent inequality for percolation β 2 δ the inequality in the title is derived for standard site percolation in any dimension assuming only that the percolation density vanishes at the critical point the proof based on a lattice animal expansion is fairly simple and is applicable to rather general site or bond short or long range independent percolation models

percolation beyond ℤd the contributions of oded schramm oded schramm 1961 2008 influenced greatly the development of percolation theory beyond the usual d setting in particular the case of nonamenable lattices here we review some of his work in this field

percolation of poisson sticks on the plane we consider a percolation model on the plane which consists of 1 dimensional sticks placed at points of a poisson process onr 2 each stick having a random but bounded length and a random direction the critical probabilities are defined with respect to the occupied clusters and vacant clusters and they are shown to be equal the equality is shown through a pivotal cell argument using a version of the russo seymour welsh theorem which we obtain for this model

a 0 1 law for the massive gaussian free field we investigate the phase transition in a non planar correlated percolation model with long range dependence obtained by considering level sets of a gaussian free field with mass above a given height h the dependence present in the model is a notorious impediment when trying to analyze the behavior near criticality alongside the critical threshold h for percolation a second parameter h ge h characterizes a strongly subcritical regime we prove that the relevant crossing probabilities converge to 1 polynomially fast below h which firmly suggests that the phase transition is sharp a key tool is the derivation of a suitable differential inequality for the free field that enables the use of a conditional influence theorem

sharpness of the phase transition for continuum percolation in mathbb r 2 r 2 we study the phase transition of random radii poisson boolean percolation around each point of a planar poisson point process we draw a disc of random radius independently for each point the behavior of this process is well understood when the radii are uniformly bounded from above in this article we investigate this process for unbounded and possibly heavy tailed radii distributions under mild assumptions on the radius distribution we show that both the vacant and occupied sets undergo a phase transition at the same critical parameter lambda c moreover the techniques we develop in this article can be applied to other models such as the poisson voronoi and confetti percolation

analyticity properties and power law estimates of functions in percolation theory we consider percolation on the sites of a graphg e g a regulard dimensional lattice all sites ofg are occupied vacant with probabilityp respectively q 1 p independently of each other w denotes the cluster of occupied sites containing a fixed site which will usually be taken to be the origin andw the cardinality ofw the percolation probability is the probability that w i e p pp w some critical values ofp ph andpt are defined respectively as the smallest value ofp for which p 0 and for which the expectation of w is infinite formally ph inf p p 0 andpt inf p ep w we show for fairly general graphsgthat ifp pt thenpp w n decreases exponentially inn for the special casesg g0 the simple quadratic lattice andg1 the graph which corresponds to bond percolation on 2 we obtain upper and lower bounds for p of the formc p ph and bounds forep w of the formc p ph we also investigate smoothness properties of p ep number of clusters per site ep w 1 w 1 this function was introduced by sykes and essam who assumed that has exactly one singularity namely atp ph for the graphsg0 andg1 i e site or bond percolation on 2 we show that p is analytic atp ph and has two continuous derivatives atp ph the emphasis is on rigorous proofs

the self dual point of the two dimensional random cluster model is critical for q 1 we prove a long standing conjecture on random cluster models namely that the critical point for such models with parameter q 1 on the square lattice is equal to the self dual point p sd q sqrt q 1 sqrt q this gives a proof that the critical temperature of the q state potts model is equal to log 1 sqrt q for all q 2 we further prove that the transition is sharp meaning that there is exponential decay of correlations in the sub critical phase the techniques of this paper are rigorous and valid for all q 1 in contrast to earlier methods valid only for certain given q the proof extends to the triangular and the hexagonal lattices as well

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

a convex parameterization of robustly stabilizing controllers this paper treats synthesis of robust controllers for linear time invariant systems uncertain real parameters are assumed to appear linearly in the closed loop characteristic polynomial the main contribution is to give a convex parameterization of all controllers that simultaneously stabilize the system for all possible parameter combinations with the new parameterization certain robust performance problems can be stated in terms of quasi convex optimization

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

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

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

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