domination by product measures we consider families of 0 1 valued random variables indexed by the vertices of countable graphs with bounded degree first we show that if these random variables satisfy the property that conditioned on what happens outside of the neighborhood of each given site the probability of seeing a 1 at this site is at least a value p which is large enough then this random field dominates a product measure with positive density moreover the density of this dominated product measure can be made arbitrarily close to 1 provided that p is close enough to 1 next we address the issue of obtaining the critical value of p defined as the threshold above which the domination by positive density product measures is assured for the graphs which have as vertices the integers and edges connecting vertices which are separated by no more than k units this critical value is shown to be 1 k k k 1 k 1 and a discontinuous transition is shown to occur similar critical values of p are found for other classes of probability measures on 0 1 z for the class of k dependent measures the critical value is again 1 k k k 1 k 1 with a discontinuous transition for the class of two block factors the critical value is shown to be 1 2 and a continuous transition is shown to take place in this case thus both the critical value and the nature of the transition are different in the two block factor and 1 dependent cases

the ring structure on the cohomology of coordinate subspace arrangements every simplicial complex delta subset2 n on the vertex set n 1 ldots n defines a real resp complex arrangement of coordinate subspaces in mathbb r n resp mathbb c n via the correspondence delta ni sigma mapsto rm span e i i in sigma the linear structure of the cohomology of the complement of such an arrangement is explicitly given in terms of the combinatorics of delta and its links by the goresky macpherson formula here we derive by combinatorial means the ring structure on the integral cohomology in terms of data of delta we provide a non trivial example of different cohomology rings in the real and complex case furthermore we give an example of a coordinate arrangement that yields non trivial multiplication of torsion elements

combinatorial stratification of complex arrangements we present a method for discretizing complex hyperplane arrange ments by encoding their topology into a finite partially ordered set of sign vectors this is used in the following ways 1 a general method is given for constructing regular cell complexes having the homotopy type of the complement of the arrangement 2 for the case of complexified arrangements this specializes to the con struction of salvetti s we study the combinatorial structure of complexified arrangements and the salvetti complex in some detail 3 this general method simultaneously produces cell decompositions of the singularity link this link is shown to have the homotopy type of a wedge of d spheres for arrangements in cd d 4 4 the homology of the link and the cohomology of the complement are computed in terms of explicit bases which are matched by alexander duality this gives a new more elementary and more generally valid proof for the brieskorn orlik solomon theorem and some related results 5 our setup leads to a more general notion of 2 pseudoarrangements which can be thought of as topologically deformed complex arrangements re taining only the essential topological and combinatorial structure we show that all of the above remains true in this generality except for the sign patterns of the orlik solomon relations department of mathematics royal institute of technology s10044 stockholm sweden e mail address andersb nada kth se institut fur mathematik universitat augsburg d 8900 augsburg germany current address institut mittag leffler auravagen 17 s 18262 djursholm sweden address after 1 april 1992 konrad zuse zentrum fur informationstechnik berlin heilbronner str 10 w 1000 berlin 31 germany e mail addresses ziegler ml kva se ziegler zib berlin de this content downloaded from 157 55 39 211 on fri 24 jun 2016 06 44 45 utc all use subject to http about jstor org terms

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

bijective linear maps on semimodules spanned by boolean matrices of fixed rank abstract let m m n b be the semimodule of all m n boolean matrices where b is the boolean algebra with two elements let k be a positive integer such that 2 k min m n let b m n k denote the subsemimodule of m m n b spanned by the set of all rank k matrices we show that if t is a bijective linear mapping on b m n k then there exist permutation matrices p and q such that t a paq for all a b m n k or m n and t a pa t q for all a b m n k this result follows from a more general theorem we prove concerning the structure of linear mappings on b m n k that preserve both the weight of each matrix and rank one matrices of weight k 2 here the weight of a boolean matrix is the number of its nonzero entries

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 note on truncated long range percolation with heavy tails on oriented graphs we consider oriented long range percolation on a graph with vertex set mathbb z d times mathbb z and directed edges of the form langle x t x y t 1 rangle for x y in mathbb z d and t in mathbb z any edge of this form is open with probability p y independently for all edges under the assumption that the values p y do not vanish at infinity we show that there is percolation even if all edges of length more than k are deleted for k large enough we also state the analogous result for a long range contact process on mathbb z d

on the truncation of systems with non summable interactions in this note we consider long range q states potts models on z d d 2 for various families of non summable ferromagnetic pair potentials x 0 we show that there exists for all inverse temperature 0 an integer n such that the truncated model in which all interactions between spins at distance larger than n are suppressed has at least q distinct infinite volume gibbs states this holds in particular for all potentials whose asymptotic behaviour is of the type x x 0 d these results are obtained using simple percolation arguments

linear decision trees subspace arrangements and mobius functions let p be a set in rn given a point x we want to test if x e p our model of computation is a linear decision tree a rooted ternary tree t where with each interior node v is associated a linear function lv x l j ajxj b and the three edges connecting v to its descendants are labelled starting from the root we move down the tree at each internal node v we check whether iv x 0 and follow the appropriately labelled edge leaves are labelled yes and no and arriving at a leaf we read off the answer to the question is x e p linear decision trees are sometimes surprisingly powerful devices we mention here the result that some np complete problems like the knapsack problem have polynomial depth linear decision trees mh lower bound results on the size or depth of linear decision trees usually depend on counting the number of connected components of p or r n p this method was introduced by dobkin and lipton dl and further developed by steele and yao sy ben or bo and others in this paper we describe topological methods that can be applied to obtain lower bounds also if these sets are connected the basic idea of this paper was first explored in bly the euler characteristic of p or r n p was used to estimate the number of leaves in a linear decision tree for p here we improve and extend the topological methods of bly in particular we show that the euler characteristic can be replaced by the sum of the betti numbers of r n p in bl y the method was applied to the case when p is a subspace arrangement i e a union of a finite number of affine subspaces the euler characteristic of the union is essentially the mobius function of the intersection lattice of these subspaces in this paper using a formula of goresky and macpherson gm we work with the betti numbers of the complement of this union and thereby obtain better bounds the motivating problem in bly was the k equal problem given n real numbers xl xn decide if some k of them are equal it was shown that ev ry linear decision tree for the k equal problem has size at least 2njk 0 n and consequently depth at least q nlog 2njk which is best possible we

on the continuity of the critical value for long range percolation in the exponential case we show that for a long range percolation model with exponentially decaying connections the limit of critical values of any sequence of long range percolation models approaching the original model from below is the critical value for the original long range percolation model as an interesting corollary this implies that if a long range percolation model with exponential connections is supercritical then it still percolates even if all long bonds are removed we also show that the percolation probability is continuous in a certain sense in the supercritical regime for long range percolation models with exponential connections

strict inequality for critical values of potts models and random cluster processes we prove that the critical value c of a ferromagnetic potts model is astrictly decreasing function of the strengths of interaction of the process this is achieved in the more general context of the random cluster representation of fortuin and kasteleyn by deriving and utilizing a formula which generalizes the technique known in percolation theory as russo s formula as a byproduct of the method we present a general argument for showing that at any given point on the critical surface of a multiparameter process the values of a certain critical exponent do not depend on the direction of approach of that point our results apply to all random cluster processes satisfying the fkg inequality

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

hyperplane arrangement cohomology and monomials in the exterior algebra we show that if x is the complement of a complex hyperplane arrangement then the homology of x has linear free resolution as a module over the exterior algebra on the first cohomology of x we study invariants of x that can be deduced from this resolution a key ingredient is a result of aramova avramov and herzog 2000 on resolutions of monomial ideals in the exterior algebra we give a new conceptual proof of this result

transience recurrence and critical behavior for long range percolation we study the behavior of the random walk on the infinite cluster of independent long range percolation in dimensions d 1 2 where x and y are connected with probability we show that if d s 2d then there is no infinite cluster at criticality this result is extended to the free random cluster model a second corollary is that when d 2 and d s 2d we can erase all long enough bonds and still have an infinite cluster the proof of recurrence in two dimensions is based on general stability results for recurrence in random electrical networks in particular we show that i i d conductances on a recurrent graph of bounded degree yield a recurrent electrical network

mean field driven first order phase transitions in systems with long range interactions we consider a class of spin systems on d with vector valued spins sx that interact via the pair potentials jx ysx sy the interactions are generally spread out in the sense that the jx y s exhibit either exponential or power law fall off under the technical condition of reflection positivity and for sufficiently spread out interactions we prove that the model exhibits a first order phase transition whenever the associated mean field theory signals such a transition as a consequence e g in dimensions d 3 we can finally provide examples of the 3 state potts model with spread out exponentially decaying interactions which undergoes a first order phase transition as the temperature varies similar transitions are established in dimensions d 1 2 for power law decaying interactions and in high dimensions for next nearest neighbor couplings in addition we also investigate the limit of infinitely spread out interactions specifically we show that once the mean field theory is in a unique state then in any sequence of translation invariant gibbs states various observables converge to their mean field values and the states themselves converge to a product measure

on cohomology algebras of complex subspace arrangements the integer cohomology algebra of the complement of a complex subspace arrangement with geometric intersection lattice is completely determined by the combinatorial data of the arrangement we give a combinatorial presentation of the cohomology algebra in the spirit of the orlik solomon result on the cohomology algebras of complex hyperplane arrangements our methods are elementary we work with simplicial models for the complements that are induced by combinatorial stratifications of complex space we describe simplicial cochains that generate the cohomology among them we distinguish a linear basis study cup product multiplication and derive an algebra presentation in terms of generators and relations

combinatorial geometries convex polyhedra and schubert cells this paper is a continuation of gm which was published in the same journal we will explore a remarkable connection between the geometry of the schubert cells in the grassmann manifold the theory of convex polyhedra and the theory of combinatorial geometries in the sense of crapo and rota cr the results in this paper were obtained simultaneously and independently by gelfand and serganova in moscow and by goresky and macpherson at the i h e s in paris as part of larger programs with different purposes see below the geometry of this simple example is so beautiful that we decided to publish it independently of the applications we believe that combinatorial methods will play an increasing role in the future of geometry and topology we consider the grassmann manifold gi of all n k dimensional subspaces of c by fixing the standard basis in c we obtain an action of the torus h i on g pk which is induced from stretching the coordinate axes in c see also sect 1 we will describe not only the trajectories but also the strata of a new and interesting decomposition of the 301 oool 8708 87 7 50

boolean rank preserving operators and boolean rank 1 spaces abstract we study the extent to which certain theorems on linear operators on field valued matrices carry over to linear operators on boolean matrices we obtain analogues and near analogues of several such theorems one of these leads us to consider linear spaces of m n boolean matrices whose nonzero members all have boolean rank 1 we obtain a structure theorem for such spaces that enables us to determine the maximum boolean dimension of such spaces and their maximum cardinality

homotopy types of subspace arrangements via diagrams of spaces we prove combinatorial formulas for the homotopy type of the union of the subspaces in an affine compactified affine spherical or projective subspace arrangement from these formulas we derive results of goresky macpherson on the homology of the arrangement and the cohomology of its complement the union of an arrangement can be interpreted as the direct limit of a diagram of spaces over the intersection poset a closely related space is obtained by taking the homotopy direct limit of this diagram our method consists in constructing a combinatorial model diagram over the same poset whose homotopy limit can be compared to the original one by usual homotopy comparison results for diagrams of spaces

k parabolic subspace arrangements in this paper we study k parabolic arrangements a generalization of the k equal arrangement for any finite real reflection group when k 2 these arrangements correspond to the well studied coxeter arrangements brieskorn 1971 showed that the fundamental group of the complement of the type w coxeter arrangement over mathbb c is isomorphic to the pure artin group of type w khovanov 1996 gave an algebraic description for the fundamental group of the complement of the 3 equal arrangement over mathbb r we generalize khovanov s result to obtain an algebraic description of the fundamental group of the complement of the 3 parabolic arrangement for arbitrary finite reflection group our description is a real analogue to brieskorn s description

truncated long range percolation on oriented graphs we consider different problems within the general theme of long range percolation on oriented graphs our aim is to settle the so called truncation question described as follows we are given probabilities that certain long range oriented bonds are open assuming that the sum of these probabilities is infinite we ask if the probability of percolation is positive when we truncate the graph disallowing bonds of range above a possibly large but finite threshold we give some conditions in which the answer is affirmative we also translate some of our results on oriented percolation to the context of a long range contact process

the low temperature phase of kac ising models we analyze the low temperature phase of ferromagnetic kax ising models in dimensionsd 2 we show that if the range of interactions is 1 then two disjoint translation invariant gibbs states exist if the inverse temperature satisfies 1 n where d 1 2d 2 d 1 for any e 0 the proof involves the blocking procedure usual for kac models and also a contour representation for the resulting long range almost continuous spin system which is suitable for the use of a variant of the peierls argument

the cohomology rings of complements of subspace arrangements the ring structure of the integral cohomology of complements of real linear subspace arrangements is considered while the additive structure of the cohomology is given in terms of the intersection poset and dimension function by a theorem of goresky and macpherson we describe the multiplicative structure in terms of the intersection poset the dimension function and orientations of the participating subspaces for the class of arrangements without pairs of intersections of codimension one in particular this yields a description of the integral cohomology ring of complex arrangements conjectured by yuzvinsky for general real arrangements a weaker result is obtained the approach is geometric and the methods are elementary

higher homotopy groups of complements of complex hyperplane arrangements abstract we generalize results of hattori on the topology of complements of hyperplane arrangements from the class of generic arrangements to the much broader class of hypersolvable arrangements we show that the higher homotopy groups of the complement vanish in a certain combinatorially determined range and we give an explicit z 1 module presentation of p the first non vanishing higher homotopy group we also give a combinatorial formula for the 1 coinvariants of p for affine line arrangements whose cones are hypersolvable we provide a minimal resolution of 2 and study some of the properties of this module for graphic arrangements associated to graphs with no 3 cycles the algorithm for computing 2 is purely combinatorial the fitting varieties associated to 2 may distinguish the homotopy 2 types of arrangement complements with the same 1 and the same betti numbers in low degrees

rank one preservers between spaces of boolean matrices in this paper we characterize i linear transformations from one space of boolean matrices to another that send pairs of distinct rank one elements to pairs of distinct rank one elements and ii surjective mappings from one space of boolean matrices to another that send rank one matrices to rank one matrices and preserve order relation in both directions both results are proved in a more general setting of tensor products of two boolean vector spaces of arbitrary dimension

non analyticity and the van der waals limit reference gr pf article 2004 001doi 10 1023 b joss 0000012506 98828 ddview record in web of science record created on 2008 11 27 modified on 2017 05 12