the connection matrix theory for morse decompositions the connection matrix theory for morse decompositions is introduced the connection matrices are matrices of maps between the homology indices of the sets in the morse decomposition the connection matrices cover in a natural way the homology index braid of the morse decomposition and provide information about the structure of the morse decomposition the existence of connection matrices of morse decompositions is established and examples illustrating applications of the connection matrix are provided

the covariance matrix of the potts model a random cluster analysis we consider the covariance matrix g mm q2 of thed dimensionalq states potts model rewriting it in the random cluster representation of fortuin and kasteleyn in any of theq ordered phases we identify the eigenvalues of this matrix both in terms of representations of the unbroken symmetry group of the model and in terms of random cluster connectivities and covariances thereby attributing algebraic significance to these stochastic geometric quantities we also show that the correlation length corresponding to the decay rate of one of the eigenvalues is the same as the inverse decay rate of the diameter of finite clusers for dimensiond 2 we show that this correlation length and the correlation length of the two point function with free boundary conditions at the corresponding dual temperature are equal up to a factor of two for systems with first order transitions this relation helps to resolve certain inconsistencies between recent exact and numerical work on correlation lengths at the self dual point o for systems with second order transitions this relation implies the equality of the correlation length exponents from above and below threshold as well as an amplitude ratio of two in the course of proving the above results we establish several properties of independent interest including left continuity of the inverse correlation length with free boundary conditions and upper semicontinuity of the decay rate for finite clusters in all dimensions and left continuity of the two dimensional free boundary condition percolation probability at o we also introduce dlr equations for the random cluster model and use them to establish ergodicity of the free measure in order to prove these results we introduce a new class of events which we call decoupling events and two inequalities for these events the first is similar to the fkg inequality but holds for events which are neither increasing nor decreasing the second is similar to the van den berg kesten inequality in standard percolation both inequalities hold for an arbitrary fkg measure

complete cohomology and gorensteinness of schemes abstract we develop and study tate and complete cohomology theory in the category of sheaves of o x modules different approaches are included we study the properties of these theories and show their power in reflecting the gorensteinness of the underlying scheme the connection of these two theories will be discussed

maximum entropy principles for disordered spins we transform nonstationary independent random fields with exponential radon nikodym factors and study the asymptotics of the transformed processes as applications we deduce conditional limit theorems for such random fields and we study a curie weiss type mean field model of a quenched mixed magnetic crystal this model has quenched site disorder and frustration but non random coupling constants we find a continuous phase transition with critical exponents equal to those of the classical curie weiss theory

almost sure quasilocality fails for the random cluster model on a tree we study the random cluster model on a homogeneous tree and show that the following three conditions are equivalent for a random cluster measure quasilocality almost sure quasilocality and the almost sure nonexistence of infinite clusters as a consequence of this we find that the plus measure for the ising model on a tree at sufficiently low temperatures can be mapped via a local stochastic transformation into a measure which fails to be almost surely quasilocal

gaussian behavior of loop erased self avoiding random walk in four dimensions une marche auto evitante de longueur n sur le reseau entier z d est une suite ordonnee de points x 0 0 x 1 x n avec x i x i 1 1 et x i x j pour i j on etudie le comportement d une telle marche a d 4 quand on applique le processus de boucle effacee

sanov results for glauber spin glass dynamics in this paper we prove a sanov result i e a large deviation principle ldp for the distribution of the empirical measure for the annealed glauber dynamics of the sherrington kirkpatrick spin glass without restrictions on time or temperature we prove a full ldp for the asymmetric dynamics and the crucial upper large deviations bound for the symmetric dynamics in the symmetric model a new order parameter arises corresponding to the response function in sozi83 in the asymmetric case we show that the corresponding rate function has a unique minimum given as the solution of a self consistent equation the key argument used in the proofs is a general result for mixing of what is known as large deviation systems lds with measures obeying an independent ldp

infinite volume limit for the stationary distribution of abelian sandpile models we study the stationary distribution of the standard abelian sandpile model in the box n n n d d for d 2 we show that as n the finite volume stationary distributions weakly converge to a translation invariant measure on allowed sandpile configurations in d this allows us to define infinite volume versions of the avalanche size distribution and related quantities the proof is based on a mapping of the sandpile model to the uniform spanning tree due to majumdar and dhar and the existence of the wired uniform spanning forest measure on d in the case d 4 we also make use of wilson s method

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

infinite clusters in percolation models the qualitative nature of infinite clusters in percolation models is investigated the results which apply to both independent and correlated percolation in any dimension concern the number and density of infinite clusters the size of their external surface the value of their total surface to volume ratio and the fluctuations in their density in particular it is shown thatn0 the number of distinct infinite clusters is either 0 1 or and the casen0 which might occur in sufficiently high dimension is analyzed

entropy for translation invariant random cluster measures we study translation invariant random cluster measures with techniques from large deviation theory and convex analysis in particular we prove a large deviation principle with rate function given by a specific entropy and a dobrushinlanfordruelle variational principle that characterizes translation invariant random cluster measures as the solutions of the variational equation for free energy consequences of these theorems include inequalities for edge and cluster densities of translationinvariant random cluster measures 1 introduction the random cluster model was introduced around 1972 by fortuin and kasteleyn 18 as a family of models that contains the bernoulli bond percolation model the ising model and the q state potts models as special cases it has developed into an important and very successful tool for studying the ising and potts models our paper focuses on the properties of the random cluster measures themselves our starting point is grimmett s 23 development of elements of a thermodynamic formalism for this model beginning with a definition of infinite volume random cluster measures in terms of a dlr dobrushinlanfordruelle equation just as for gibbsian systems natural next questions concern the validity of large deviation principles and variational characterizations of infinite volume measures these questions we answer in the present paper and derive some consequences here is a brief overview of the dlr approach of 23 the random cluster measures of interest live on the graph formed by putting edges between nearest neighbor vertices of the d dimensional hypercubic lattice d the edge that connects two vertices x and y that are l 1 distance one apart is

large deviations for the empirical field of a gibbs measure on applique le principe de contraction pour obtenir un principe de grande deviation pour la distribution des distributions empiriques on donne une description detaillee de la fonction de taux resultante

uniqueness of the infinite component in a random graph with applications to percolation and spin glasses we extend the theorem of burton and keane on uniqueness of the infinite component in dependent percolation to cover random graphs on d or d with long range edges we also study a short range percolation model related to nearest neighbor spin glasses on d or on a slab d 0 k and prove both that percolation occurs and that the infinite component is unique forv 2 0 1 or larger

abelian sandpiles and the harmonic model we present a construction of an entropy preserving equivariant surjective map from the d dimensional critical sandpile model to a certain closed shift invariant subgroup of mathbb t mathbb z d the harmonic model a similar map is constructed for the dissipative abelian sandpile model and is used to prove uniqueness and the bernoulli property of the measure of maximal entropy for that model

the stochastic random cluster process and the uniqueness of random cluster measures the random cluster model is a generalization of percolation and ferromagnetic potts models due to fortuin and kasteleyn not only is the random cluster model a worthwhile topic for study in its own right but also it provides much information about phase transitions in the associated physical models this paper serves two functions first we introduce and survey random cluster measures from the probabilist s point of view giving clear statements of some of the many open problems second we present new results for such measures as follows we discuss the relationship between weak limits of random cluster measures and measures satisfying a suitable dlr condition using an argument based on the convexity of pressure we prove the uniqueness of random cluster measures for all but at most countably many values of the parameter p related results concerning phase transition in two or more dimensions are included together with various stimulating conjectures the uniqueness of the infinite cluster is employed in an intrinsic way in part of these arguments in the second part of this paper is constructed a markov process whose level sets are reversible markov processes with random cluster measures as unique equilibrium measures this construction enables a coupling of random cluster measures for all values of p furthermore it leads to a proof of the semicontinuity of the percolation probability and provides a heuristic probabilistic justification for the widely held belief that there is a first order phase transition if and only if the cluster weighting factor q is sufficiently large

theory of minimum spanning trees i mean field theory and strongly disordered spin glass model the minimum spanning tree mst is a combinatorial optimization problem given a connected graph with a real weight cost on each edge find the spanning tree that minimizes the sum of the total cost of the occupied edges we consider the random mst in which the edge costs are quenched independent random variables there is a strongly disordered spin glass model due to newman and stein phys rev lett 72 2286 1994 which maps precisely onto the random mst we study scaling properties of random msts using a relation between kruskal s greedy algorithm for finding the mst and bond percolation we solve the random mst problem on the bethe lattice bl with appropriate wired boundary conditions and calculate the fractal dimension d 6 of the connected components viewed as a mean field theory the result implies that on a lattice in euclidean space of dimension d there are of order w d ensuremath d large connected components of the random mst inside a window of size w and that d d c d 6 is a critical dimension this differs from the value 8 suggested by newman and stein we also critique the original argument for 8 and provide an improved scaling argument that again yields d c 6 the result implies that the strongly disordered spin glass model has many ground states for dg6 and only of order one below six the results for msts also apply on the poisson weighted infinite tree which is a mean field approach to the continuum model of msts in euclidean space and is a limit of the bl in a companion paper we develop an ensuremath epsilon 6 ensuremath d expansion for the random mst on critical percolation clusters

large deviations for gibbs random fields a large deviation principle for gibbs random fields on zd is proven and a corresponding large deviations proof of the gibbs variational formula is given a generalization of the lanford theory of large deviations is also obtained

computational methods for bifurcation problems with symmetries with special attention to steady state and hopf bifurcation points we show how group theoretical methods can be employed to utilize the symmetry of a bifurcation problem in numerical computations we extend the approach by werner 1988 by presenting methods for the detection of bifurcation points and the computation of multiple hopf points the essential numerical point is the utilization of certain reduced instead of full systems involving appropriate subgroups of the underlying symmetry group the group theoretical tool is an a priori knowledge of the interaction of certain subgroups 0 and of at possibly multiple steady state or hopf bifurcation points minimal 0 breaking bifurcation we introduce a bifurcation graph which shows graphically this a priori information its edges represent possible symmetry breaking bifurcations our analysis follows the lines of golubitsky stewart and schaeffer 1988 but it is aimed to numerical applications we have chosen a 4 box brusselator model in order to explain our notions and ideas and to discuss the numerical procedure

entropy limit theorems and variational principles for disordered lattice systems we study infinite volume limits and gibbs states of disordered lattice systems with bounded and continuous potentials our main tools are a generalization of relative entropy for random reference measures and a large deviation theory for nonstationary independent processes we find that many familiar results of invariant potentials such as large deviation theorems variational principles and equivalence of ensembles continue to hold for disordered models with suitably modified statements

the component graph of the uniform spanning forest transitions in dimensions 9 10 11 ldots 9 10 11 we prove that the uniform spanning forests of mathbb z d and mathbb z ell have qualitatively different connectivity properties whenever ell d ge 4 in particular we consider the graph formed by contracting each tree of the uniform spanning forest down to a single vertex which we call the component graph we introduce the notion of ubiquitous subgraphs and show that the set of ubiquitous subgraphs of the component graph changes whenever the dimension changes and is above 8 to separate dimensions 5 6 7 and 8 we prove a similar result concerning ubiquitous subhypergraphs in the component hypergraph our result sharpens a theorem of benjamini kesten peres and schramm who proved that the diameter of the component graph increases by one every time the dimension increases by four

the random cluster model on the complete graph the random cluster model of fortuin and kasteleyn contains as special cases the percolation ising and potts models of statistical physics when the underlying graph is the complete graph onn vertices then the associated processes are called mean field in this study of the mean field random cluster model with parametersp n andq we show that its properties for any value ofq 0 may be derived from those of an erd s renyi random graph in this way we calculate the critical point c q of the model and show that the associated phase transition is continuous if and only ifq 2 exact formulae are given for c q the density of the largest component the density of edges of the model and the free energy this work generalizes earlier results valid for the potts model whereq is an integer satisfyingq 2 equivalent results are obtained for a fixed edge number random cluster model as a consequence of the results of this paper one obtains large deviation theorems for the number of components in the classical random graph models whereq 1

geometry of the uniform spanning forest transitions in dimensions 4 8 12 the uniform spanning forest usf in d is the weak limit of random uniformly chosen spanning trees in n n d pemantle 11 proved that the usf consists a s of a single tree if and only if d 4 we prove that any two components of the usf in d are adjacent a s if 5 d 8 but not if d 9 more generally let n x y be the minimum number of edges outside the usf in a path joining x and y in d then max n x y x y in mathbb z d left lfloor d 1 4 right rfloor textrm a s

global specifications and nonquasilocality of projections of gibbs measures reference gr pf article 1997 001doi 10 1214 aop 1024404514view record in web of science record created on 2008 11 27 modified on 2017 05 12

local characteristics entropy and limit theorems for spanning trees and domino tilings via transfer impedances let g be a finite graph or an infinite graph on which mathbb z d acts with finite fundamental domain if g is finite let mathbf t be a random spanning tree chosen uniformly from all spanning trees of g if g is infinite methods from pemantle show that this still makes sense producing a random essential spanning forest of g a method for calculating local characteristics i e finite dimensional marginals of mathbf t from the transfer impedance matrix is presented this differs from the classical matrix tree theorem in that only small pieces of the matrix n dimensional minors are needed to compute small n dimensional marginals calculation of the matrix entries relies on the calculation of the green s function for g which is not a local calculation however it is shown how the calculation of the green s function may be reduced to a finite computation in the case when g is an infinite graph admitting a z d action with finite quotient the same computation also gives the entropy of the law of mathbf t these results are applied to the problem of tiling certain lattices by dominos the so called dimer problem another application of these results is to prove modified versions of conjectures of aldous on the limiting distribution of degrees of a vertex and on the local structure near a vertex of a uniform random spanning tree in a lattice whose dimension is going to infinity included is a generalization of moments to tree valued random variables and criteria for these generalized moments to determine a distribution

symmetry breaking for solutions of semilinear elliptic equations with general boundary conditions we study the bifurcation of radially symmetric solutions of f u 0 onn balls into asymmetric ones we show that ifu satisfies homogeneous neumann boundary conditions the asymmetric components in the kernel of the linearized operators can have arbitrarily high dimension for general boundary conditions we prove some theorems which give bounds on the dimensions of the set of asymmetric solutions and on the structure of the kernels of the linearized operators

absolute relative and tate cohomology of modules of finite gorenstein dimension we study finitely generated modules m over a ring r noetherian on both sides if m has finite gorenstein dimension mbox g dim rm in the sense of auslander and bridger then it determines two other cohomology theories besides the one given by the absolute cohomology functors rm ext n r m relative cohomology functors rm ext n mathcal g m are defined for all non negative integers n they treat the modules of gorenstein dimension 0 as projectives and vanish for n mbox g dim rm tate cohomology functors widehat rm ext n r m are defined for all integers n all groups widehat rm ext n r m n vanish if m or n has finite projective dimension comparison morphisms varepsilon mathcal g n colon rm ext n mathcal g m to rm ext n r m and varepsilon r n colon rm ext n r m to widehat rm ext n r m link these functors we give a self contained treatment of modules of finite g dimension establish basic properties of relative and tate cohomology and embed the comparison morphisms into a canonical long exact sequence 0 to rm ext 1 mathcal g m to cdots to rm ext n mathcal g m to rm ext n r m to widehat rm ext n r m to rm ext n 1 mathcal g m to cdots we show that these results provide efficient tools for computing old and new numerical invariants of modules over commutative local rings 2000 mathematical subject classification 16e05 13h10 18g25

correction added in proof we extend sanov s theorem on i i d large deviations to independent but not identically distributed random variables and study the generalization of relative entropy that appears as the rate function

connected simple systems and the conley index of isolated invariant sets the object of this paper is to present new and simplified proofs for most of the basic results in the index theory for flows simple explicit formulae are derived for the maps which play a central role in the theory the presentation is self contained

nonlinear analogs of linear group actions on spheres introduction this article is based upon a principle which is so standard that it is almost a cliche the first step to understanding a nonlinear phenomenon is to define and study a suitable linear approximation to be more specific we shall describe some applications of this to the symmetry questions of topological transformation groups given a topological space x let homeo denote the set of self homeomorphisms of x this is a group under composition of mappings if g is an arbitrary group then a group action of g on i is a homomorphism homeo x frequently we wish to impose some weak assumptions on p for example if g is a topological group then we might want cp to have suitable continuity properties the usual assumption is that the map

chromatic polynomials potts models and all that the q state potts model can be defined on an arbitrary finite graph and its partition function encodes much important information about that graph including its chromatic polynomial flow polynomial and reliability polynomial the complex zeros of the potts partition function are of interest both to statistical mechanicians and to combinatorists i give a pedagogical introduction to all these problems and then sketch two recent results a construction of a countable family of planar graphs whose chromatic zeros are dense in the whole complex q plane except possibly for the disc q 1 1 b proof of a universal upper bound on the q plane zeros of the chromatic polynomial or antiferromagnetic potts model partition function in terms of the graph s maximum degree

microcanonical distributions for lattice gases in this article a large deviation principle cf theorem 1 3 for the empirical distribution functional is applied to prove a rather general version of boltzmann s principle cf theorem 3 5 for models with shift invariant finite range potentials the final section contains an application of these considerations to the two dimensional ising model at sub critical temperature

discrete systems and abelian sandpiles abstract computations on the sandpile model lead to questions concerning discrete behaviors it is shown that the original question has an answer which is not helpful for the study of sandpiles a variation of the question using periodic solutions leads to better answers these might have an interpretation for the study of sandpiles

surface order large deviations for ising potts and percolation models we derive uniform surface order large deviation estimates for the block magnetization in finite volume ising or potts models with plus or free or a combination of both boundary conditions in the phase coexistence regime ford 3 the results are valid up to a limit of slab thresholds conjectured to agree with the critical temperature our arguments are based on the renormalization of the random cluster model withq 1 andd 3 and on corresponding large deviation estimates for the occurrence in a box of a largest cluster with density close to the percolation probability the results are new even for the case of independent percolation q 1 as a byproduct of our methods we obtain further results in the fk model concerning semicontinuity inp andq of the percolation probability the second largest cluster in a box and the tail of the finite cluster size distribution

density and uniqueness in percolation two results on site percolation on thed dimensional lattice d 1 arbitrary are presented in the first theorem we show that for stationary underlying probability measures each infinite cluster has a well defined density with probability one the second theorem states that if in addition the probability measure satisfies the finite energy condition of newman and schulman then there can be at most one infinite cluster with probability one the simple arguments extend to a broad class of finite dimensional models including bond percolation and regular lattices

minimal spanning forests minimal spanning forests on infinite graphs are weak limits of minimal spanning trees from finite subgraphs these limits can be taken with free or wired boundary conditions and are denoted fmsf free minimal spanning forest and wmsf wired minimal spanning forest respectively the wmsf is also the union of the trees that arise from invasion percolation started at all vertices we show that on any cayley graph where critical percolation has no infinite clusters all the component trees in the wmsf have one end a s in z d this was proved by alexander ann probab 23 1995 87 104 but a different method is needed for the nonamenable case we also prove that the wmsf components are thin in a different sense namely on any graph each component tree in the wmsf has p c 1 a s where p c denotes the critical probability for having an infinite cluster in bernoulli percolation on the other hand the fmsf is shown to be thick on any connected graph the union of the fmsf and independent bernoulli percolation with arbitrarily small parameter is a s connected in conjunction with a recent result of gaboriau this implies that in any cayley graph the expected degree of the fmsf is at least the expected degree of the fsf the weak limit of uniform spanning trees we also show that the number of infinite clusters for bernoulli p u percolation is at most the number of components of the fmsf where p u denotes the critical probability for having a unique infinite cluster finally an example is given to show that the minimal spanning tree measure does not have negative associations

choosing a spanning tree for the integer lattice uniformly consider the nearest neighbor graph for the integer lattice zd in d dimensions for a large finite piece of it consider choosing a spanning tree for that piece uniformly among all possible subgraphs that are spanning trees as the piece gets larger this approaches a limiting measure on the set of spanning graphs for zd this is shown to be a tree if and only if d 4 in this case the tree has only one topological end that is there are no doubly infinite paths when d 2 5 the spanning forest has infinitely many components almost surely with each component having one or two topological ends

cohomology theories based on gorenstein injective modules in this paper we study relative and tate cohomology of modules of finite gorenstein injective dimension using these cohomology theories we present variations of grothendieck local cohomology modules namely gorenstein and tate local cohomology modules by applying a sort of avramov martsinkovsky exact sequence we show that these two variations of local cohomology are tightly connected to the generalized local cohomology modules introduced by j herzog we discuss some properties of these modules and give some results concerning their vanishing and non vanishing

uniform spanning forests we study uniform spanning forest measures on infinite graphs which are weak limits of uniform spanning tree measures from finite subgraphs these limits can be taken with free fsf or wired wsf boundary conditions pemantle proved that the free and wired spanning forests coincide in z d and that they give a single tree iff d 4 in the present work we extend pemantle s alternative to general graphs and exhibit further connections of uniform spanning forests to random walks potential theory invariant percolation and amenability the uniform spanning forest model is related to random cluster models in statistical physics but because of the preceding connections its analysis can be carried further among our results are the following the fsf and wsf in a graph g coincide iff all harmonic dirichlet functions on g are constant the tail fields of the wsf and the fsf are trivial on any graph on any cayley graph that is not a finite extension of z all component trees of the wsf have one end this is new in z d for d 5 on any tree as well as on any graph with spectral radius less than 1 a s all components of the wsf are recurrent the basic topology of the free and the wired uniform spanning forest measures on lattices in hyperbolic space h d is analyzed a cayley graph is amenable iff for all e 0 the union of the wsf and bernoulli percolation with parameter e is connected harmonic measure from infinity is shown to exist on any recurrent proper planar graph with finite codegrees we also present numerous open problems and conjectures

large deviations for lattice systems i parametrized independent fields we prove large deviation theorems for empirical measures of independent random fields whose distributions depend measurably on an auxiliary parameter this dependence respects the action of the shift group and a large deviation principle holds whenever a certain ergodicity condition is satisfied we also investigate the entropy functions for these processes especially in relation to the usual relative entropy

indistinguishability of trees in uniform spanning forests we prove that in both the free and the wired uniform spanning forest fusf and wusf of any unimodular random rooted network in particular of any cayley graph it is impossible to distinguish the connected components of the forest from each other by invariantly defined graph properties almost surely this confirms a conjecture of benjamini et al ann probab 29 1 1 65 2001 we also answer positively two additional questions of benjamini et al ann probab 29 1 1 65 2001 under the assumption of unimodularity we prove that on any unimodular random rooted network the fusf is either connected or has infinitely many connected components almost surely and if the fusf and wusf are distinct then every component of the fusf is transient and infinitely ended almost surely all of these results are new even for cayley graphs

long range properties of spanning trees we compute some large scale properties of the uniform spanning tree process on bounded regions in z2 in particular we compute the distribution of the meeting point of the branches of the tree issued from three boundary points we also compute the crossing probabilities of branches of the tree on rectangular and annular regions as well as the winding number of the branches of the tree

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

large deviation principles for euclidean functionals and other nearly additive processes we prove a large deviation principle for a process indexed by cubes of the multidimensional integer lattice or euclidean space under approximate additivity and regularity hypotheses the rate function is the convex dual of the limiting logarithmic moment generating function in some applications the rate function can be expressed in terms of relative entropy the general result applies to processes in euclidean combinatorial optimization statistical mechanics and computational geometry examples include the length of the minimal tour the traveling salesman problem the length of the minimal matching graph the length of the minimal spanning tree the length of the k nearest neighbors graph and the free energy of a short range spin glass model

indistinguishability of the components of random spanning forests we prove that the infinite components of the free uniform spanning forest fusf of a cayley graph are indistinguishable by any invariant property given that the forest is different from its wired counterpart similar result is obtained for the free minimal spanning forest fmsf we also show that with the above assumptions there can only be 0 1 or infinitely many components which solves the problem for the fusf of caylay graphs completely these answer questions by benjamini lyons peres and schramm for cayley graphs which have been open up to now our methods apply to a more general class of percolations those satisfying weak insertion tolerance and work beyond cayley graphs in the more general setting of unimodular random graphs

subsequential scaling limits of simple random walk on the two dimensional uniform spanning tree the first main result of this paper is that the law of the rescaled two dimensional uniform spanning tree is tight in a space whose elements are measured rooted real trees continuously embedded into euclidean space various properties of the intrinsic metrics measures and embeddings of the subsequential limits in this space are obtained with it being proved in particular that the hausdorff dimension of any limit in its intrinsic metric is almost surely equal to 8 5 in addition the tightness result is applied to deduce that the annealed law of the simple random walk on the two dimensional uniform spanning tree is tight under a suitable rescaling for the limiting processes which are diffusions on random real trees embedded into euclidean space detailed transition density estimates are derived

harmonic models and spanning forests of residually finite groups abstract we prove a number of identities relating the sofic entropy of a certain class of non expansive algebraic dynamical systems the sofic entropy of the wired spanning forest and the tree entropy of cayley graphs of residually finite groups we also show that homoclinic points and periodic points in harmonic models are dense under general conditions

approach to criticality in sandpiles a popular theory of self organized criticality predicts that the stationary density of the abelian sandpile model equals the threshold density of the corresponding fixed energy sandpile we recently announced that this density conjecture is false when the underlying graph is any of z2 the complete graph kn the cayley tree the ladder graph the bracelet graph or the flower graph in this paper we substantiate this claim by rigorous proof and extensive simulations we show that driven dissipative sandpiles continue to evolve even after a constant fraction of the sand has been lost at the sink nevertheless we do find and prove a relationship between the two models the threshold density of the fixed energy sandpile is the point at which the driven dissipative sandpile begins to lose a macroscopic amount of sand to the sink

symmetry breaking for positive solutions of semilinear elliptic equations in a recent interesting paper gidas ni and nirenberg 2 proved that positive solutions of the dirichlet problem for second order semi linear elliptic equations on balls must themselves be spherically symmetric functions here we consider the bifurcation problem for such solutions specifically we investigate the ways in which the symmetric solution can bifurcate into a nonsymmetric solution when this happens we say that the symmetry breaks to carry out this program we rely on certain results in 6 where we studied the kernel of the associated linearized operator this enables us to give some necessary conditions for symmetry to break we also find a class of functionsf where these conditions are also sufficient see equation 18 below the problem is of course to show that when zero comes into the spectrum of the linearized operator and our conditions are fulfilled bifurcation into the non radial direction actually occurs this is done by showing first that the bifurcation curve is a smooth manifold near the bifurcation point and then appealing to a theorem of vanderbauwhede 8 which gives sufficient conditions for bifurcation to occur in the presence of symmetries

transfer current and pattern fields in spanning trees when a simply connected domain d subset mathbb r d d ge 2 is approximated in a good way by embedded connected weighted graphs we prove that the transfer current matrix defined on the edges of the graph viewed as an electrical network converges up to a local weight factor to the differential of green s function on d this observation implies that properly rescaled correlations of the spanning tree model and correlations of minimal subconfigurations in the abelian sandpile model have a universal and conformally covariant limit we further show that on a periodic approximation of the domain all pattern fields of the spanning tree model as well as the minimal pattern e g zero height fields of the sandpile converge weakly in distribution to gaussian white noise

regularity properties and pathologies of position space renormalization group transformations scope and limitations of gibbsian theory we reconsider the conceptual foundations of the renormalization group rg formalism and prove some rigorous theorems on the regularity properties and possible pathologies of the rg map our main results apply to local in position space rg maps acting on systems of bounded spins compact single spin space regarding regularity we show that the rg map defined on a suitable space of interactions formal hamiltonians is always single valued and lipschitz continuous on its domain of definition this rules out a recently proposed scenario for the rg description of first order phase transitions on the pathological side we make rigorous some arguments of griffiths pearce and israel and prove in several cases that the renormalized measure is not a gibbs measure for any reasonable interaction this means that the rg map is ill defined and that the conventional rg description of first order phase transitions is not universally valid for decimation or kadanoff transformations applied to the ising model in dimension d 3 these pathologies occur in a full neighborhood 0 h e of the low temperature part of the first order phase transition surface for block averaging transformations applied to the ising model in dimension d 2 the pathologies occur at low temperatures for arbitrary magnetic field strength pathologies may also occur in the critical region for ising models in dimension d 4 we discuss the heuristic and numerical evidence on rg pathologies in the light of our rigorous theorems in addition we discuss critically the concept of gibbs measure which is at the heart of present day classical statistical mechanics we provide a careful and we hope pedagogical overview of the theory of gibbsian measures as well as the less familiar non gibbsian measures emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations we give a rather complete catalogue of the known examples of such occurrences the main message of this paper is that despite a well established tradition gibbsiannessshould not be taken for granted

interfaces in the potts model i pirogov sinai theory of the fortuin kasteleyn representation we develop a new analysis of the order disorder transition in ferromagnetic potts models for large numberq of spin states we use the pirogov sinai theory which we adapt to the fortuin kasteleyn representation of the models this theory applies in a rather direct way in our approach and leads to a system of non interacting contours with small activities as a consequence simpler and more natural techniques are found allowing us to recover previous results on the bulk properties of the model which then extend to non integer values ofq and to deal with non translation invariant boundary conditions this will be applied in a second part of this work to study the behaviour of the interfaces at the transition point

sheaf cohomology and free resolutions over exterior algebras in this paper we derive an explicit version of the bernstein gel fand gel fand bgg correspondence between bounded complexes of coherent sheaves on projective space and minimal doubly infinite free reso lutions over its koszul dual exterior algebra among the facts about the bgg correspondence that we derive is that taking homology of a complex of sheaves corresponds to taking the linear part of a resolution over the exterior algebra we explore the structure of free resolutions over an exterior algebra for example we show that such resolutions are eventually dominated by their linear parts in the sense that erasing all terms of degree 1 in the complex yields a new complex which is eventually exact as applications we give a construction of the beilinson monad which expresses a sheaf on projective space in terms of its cohomology by using sheaves of differential forms the explicitness of our version allows us to to prove two conjectures about the morphisms in the monad and we get an efficient method for machine computation of the cohomology ofsheaves we also construct all the monads for a sheaf that can be built from sums of line bundles and show that they are often characterized by numerical data let v be a finite dimensional vector space over a field k and let w v be the dual space in this paper we will study complexes and resolutions over the exterior algebra e v and their relation to modules over s sym w and sheaves on projective space p w in this paper we study the bernstein gel fand gel fand bgg correspondence 1978 usually stated as an equivalence between the derived category of bounded complexes of coherent sheaves on p w and the stable category of finitely generated graded modules over e its essential content is a functor r from complexes of graded s modules to complexes of graded e modules and its adjoint l for example if m imi is a graded s module regarded as a complex with just one term then the first and third authors are grateful to the nsf for partial support during the preparation of this paper the third author wishes to thank msri for its hospitality

stable cohomology over local rings abstract for a commutative noetherian ring r with residue field k stable cohomology modules ext r n k k have been defined for each n z but their meaning has remained elusive it is proved that the k rank of any ext r n k k characterizes important properties of r such as being regular complete intersection or gorenstein these numerical characterizations are based on results concerning the structure of z graded k algebra carried by stable cohomology it is shown that in many cases it is determined by absolute cohomology through a canonical homomorphism of algebras ext r k k ext r k k some techniques developed in the paper are applicable to the study of stable cohomology functors over general associative rings

almost sure quasilocality in the random cluster model keywords random cluster model non gibbs states quasilocality of conditional distributions reference gr pf article 1995 005doi 10 1007 bf02184883 record created on 2008 11 27 modified on 2017 05 12

large deviation estimates for a conditional probability distribution applications to random interaction gibbs measures let x i y i d be independent identically distributed random variables with arbitrary distribution we show that for almost every y i i the conditional law of the empirical field given y i i satisfies to large deviation inequalities this applies to the study of gibbs measures with random interaction in the case of some mean field models as well as of short range summable interaction we show that the pressure is nonrandom and is given by a variational formula these random gibbs measures have the same large deviation rate which does not depend on the particular realization of the interaction their local behaviour is described in terms of conditional probabilities given the interaction of solutions to the variational formula

algebraic transition matrices in the conley index theory we introduce the concept of an algebraic transition matrix these are degree zero isomorphisms which are upper triangular with respect to a partial order it is shown that all connection matrices of a morse decomposition for which the partial order is a series parallel admissible order are related via a conjugation with one of these transition matrices this result is then restated in the form of an existence theorem for global bifurcations simple exarnples of how these results can be applied are also presented

variational formulas and cocycle solutions for directed polymer and percolation models we discuss variational formulas for the law of large numbers limits of certain models of motion in a random medium namely the limiting time constant for last passage percolation and the limiting free energy for directed polymers the results are valid for models in arbitrary dimension steps of the admissible paths can be general the environment process is ergodic under spatial translations and the potential accumulated along a path can depend on the environment and the next step of the path the variational formulas come in two types one minimizes over gradient like cocycles and another one maximizes over invariant measures on the space of environments and paths minimizing cocycles can be obtained from busemann functions when these can be proved to exist the results are illustrated through 1 1 dimensional exactly solvable examples periodic examples and polymers in weak disorder

pattern selection with o 3 symmetry abstract for each irreducible representation of so 3 and o 3 we determine up to conjugacy all isotropy subgroups and identify in particular the maximal isotropy subgroups each isotropy subgroup corresponds to a possible planform for the spherical benard problem using an equivariant branching lemma of cicogna 1 we prove for each of these representations the existence of solutions corresponding to a number of different planforms thus extending substantially the work of busse 2 3 and sattinger 4 we also give a useful criterion for showing when solutions obtained by the equivariant branching lemma must be unstable