percolation on infinite graphs and isoperimetric inequalities we consider the bernoulli bond percolation process with parameter p on infinite graphs and we give a general criterion for bounded degree graphs to exhibit a non trivial percolation threshold based either on a single isoperimetric inequality if the graph has a bi infinite geodesic or two isoperimetric inequalities if the graph has not a bi infinite geodesic this new criterion extends previous criteria and brings together a large class of amenable graphs such as regular lattices and non amenable graphs such trees we also study the finite connectivity in graphs satisfying the new general criterion and show that graphs in this class with a bi infinite geodesic always have finite connectivity functions with exponential decay when p is sufficiently close to one on the other hand we show that there are graphs in the same class with no bi infinite geodesic for which the finite connectivity decays sub exponentially down to polynomially in the highly supercritical phase even for p arbitrarily close to one

multivalued periodic lienard systems abstract we consider a nonlinear multivalued periodic lienard system driven by a general strictly monotone nonhomogeneous homeomorphism which includes as a special case the vector p laplacian the reaction has also a maximal monotone map which need not be defined on all of r n we prove existence theorems for both the convex and nonconvex problems we also show the existence of extremal trajectories that is solutions moving through the extremal points of the multivalued perturbation we also show that every solution of the convexified system can be obtained as the c 1 t r n limit of a sequence of certain extremal trajectories finally as an illustration we examine a nonlinear control system with a priori feedback

the alexander orbach conjecture holds in high dimensions we examine the incipient infinite cluster iic of critical percolation in regimes where mean field behavior has been established namely when the dimension d is large enough or when d 6 and the lattice is sufficiently spread out we find that random walk on the iic exhibits anomalous diffusion with the spectral dimension d s frac 4 3 that is p t x x t 2 3 o 1 this establishes a conjecture of alexander and orbach j phys lett paris 43 625 631 1982 en route we calculate the one arm exponent with respect to the intrinsic distance

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

tree graph inequalities and critical behavior in percolation models various inequalities are derived and used for the study of the critical behavior in independent percolation models in particular we consider the critical exponent associated with the expected cluster sizex and the structure of then site connection probabilities n x1 xn it is shown that quite generally 1 the upper critical dimension above which attains the bethe lattice value 1 is characterized both in terms of the geometry of incipient clusters and a diagramatic convergence condition for homogeneousd dimensional lattices with x y o x y d 2 atp p c our criterion shows that 1 if 6 d 3 the connectivity functions n are generally bounded by tree diagrams which involve the two point function we conjecture that above the critical dimension the asymptotic behavior of n in the critical regime is actually given by such tree diagrams modified by a nonsingular vertex factor other results deal with the exponential decay of the cluster size distribution and the function 2 x y

the branching random walk and contact process on galton watson and nonhomogeneous trees we show that the branching random walk on a galton watson tree may have one or two phase transitions depending on the relative sizes of the mean degree and the maximum degree we show that there are some galton watson trees on which the branching random walk has one phase transition while the contact process has two this contradicts a conjecture of madras and schinazi we show that the contact process has only one phase transition on some trees of uniformly exponential growth and bounded degree contradicting a conjecture of pemantle

solvability of strongly nonlinear boundary value problems for second order differential inclusions abstract in this paper we study a problem for a second order differential inclusion with dirichlet neumann and mixed boundary conditions the equation is driven by a nonlinear not necessarily homogeneous differential operator satisfying certain conditions and containing as a particular case the p laplacian operator we prove the existence of solutions both for the case in which the multivalued nonlinearity has convex values and for the case in which it has not convex values the presence of a maximal monotone operator in the equation make the results applicable to gradient systems with non smooth time invariant convex potential and differential variational inequalities

ising models on hyperbolic graphs we consider ising models with ferromagnetic interactions and zero external magnetic field on the hyperbolic graph v f where v is the number of neighbors of each vertex and f is the number of sides of each face let tc be the critical temperature and t c sup t tc f 2 where f is the free boundary condition b c gibbs state is the plus b c gibbs state and is the minus b c gibbs state we prove that if the hyperbolic graph is self dual i e v f or if v is sufficiently large how large depends on f e g v 35 suffices for any f 3 and v 17 suffices for any f 17 then 0 t c tc in contrast with that t c tc for ising models on the hypercubic lattice zd with d 2 a result due to lebowitz 22 while whenever t t c f 2 the last result is an improvement in comparison with the analogous statement in refs 28 and 33 in which it was only proved that f 2 when t t c and it remains to show in both papers that f 2 whenever t t c therefore t c and tc divide 0 into three intervals 0 t c t c tc and tc in which but f 2 and f 2 and respectively

infinite graphs with a nontrivial bond percolation threshold some sufficient conditions we consider bernoulli bond percolation on infinite graphs and we identify a class of graphs for which the critical percolation probability is strictly less than 1 the graphs in this class have to fulfill conditions stated in terms of a minimal cut set property and a logarithmic isoperimetric inequality for the particular case of planar graphs the condition on minimal cut sets can be be replaced by the assumption that the dual of the graph is bounded degree

generalization of fredholm alternative for nonlinear differential operators on considere le probleme aux valeurs limites u t p 2 u t f t u t g t t 0 u 0 u 0 on demontre que si lim u inf f t u u p 2 u lim u sup f t u u p 2 u sont entre des valeurs propres du probleme aux valeurs propres u p 2 u u p 2 u sur 0 u 0 u 0 alors le probleme aux limites est resoluble pour tout g l 1

indistinguishability of percolation clusters we show that when percolation produces infinitely many infinite clusters on a cayley graph one cannot distinguish the clusters from each other by any invariantly defined property this implies that uniqueness of the infinite cluster is equivalent to nondecay of connectivity a k a long range order we then derive applications concerning uniqueness in kazhdan groups and in wreath products and inequalities for p u

infinite clusters in dependent automorphism invariant percolation on trees we study dependent bond percolation on the homogeneous tree t n of order n 2 under the assumption of automorphism invariance excluding a trivial case we find that the number of infinite clusters a s is either 0 or furthermore each infinite cluster a s has either 1 2 or infinitely many topological ends and infinite clusters with infinitely many topological ends have a s a branching number greater than 1 we also show that if the marginal probability that a single edge is open is at least 2 n 1 then the existence of infinite clusters has to have positive probability several concrete examples are considered

random walks capacity and percolation on trees a collection of several different probabilistic processes involving trees is shown to have an unexpected unity this makes possible a fruitful interplay of these probabilistic processes the processes are allowed to have arbitrary parameters and the trees are allowed to be arbitrary as well our work has five specific aims first an exact correspondence between random walks and percolation on trees is proved extending and sharpening previous work of the author this is achieved by establishing surprisingly close inequalities between the crossing probabilities of the two processes second we give an equivalent formulation of these inequalities which uses a capacity with respect to a kernel defined by the percolation this capacitary formulation extends and sharpens work of fan on random interval coverings third we show how this formulation also applies to generalize work of evans on random labelling of trees fourth the correspondence between random walks and percolation is used to decide whether certain random walks on random trees are transient or recurrent a s in particular we resolve a conjecture of griffeath on the necessity of the nash williams criterion fifth for this last purpose we establish several new basic results on branching processes in varying environments

extremal solutions and strong relaxation for nonlinear multivalued systems with maximal monotone terms abstract we consider differential systems in r n driven by a nonlinear nonhomogeneous second order differential operator a maximal monotone term and a multivalued perturbation f t u u for periodic systems we prove the existence of extremal trajectories that is solutions of the system in which f t u u is replaced by ext f t u u the extreme points of f t u u for dirichlet systems we show that the extremal trajectories approximate the solutions of the convex problem in the c 1 t r n norm strong relaxation

solutions and multiple solutions for periodic systems with nonhomogeneous differential operators abstract in this paper we consider nonlinear periodic systems driven by the nonhomogeneous vector p laplacian differential operator and with a nonsmooth potential using variational techniques based on the nonsmooth critical point theory we prove existence and multiplicity results finally we obtain an existence result for systems driven by the ordinary vector p laplacian and we show that it is still true in the more general framework of the previous results

nonlinear second order differential inclusions with a series of convex functions abstract in this paper we study a class of second order differential inclusions attached to which the convex potential varies depending on the time and the boundary conditions are described by the subdifferentials of convex functions applying the theory of lower semicontinuous convex and proper functionals and fixed points we give some results about the representation of the subdifferential operators of some convex functionals and the existence of solutions for the boundary value problems

the triangle condition for contact processes on homogeneous trees we complete work of c c wu by showing that for contact processes on homogeneous trees with degree at least 3 the triangle condition is satisfied below the second critical point in particular it holds at the first critical point and therefore at this critical point the contact process has mean field critical exponents

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

ising models on the lobachevsky plane we consider the ising model on a lattice which is the orbit of a discrete cocompact group acting on the hyperbolic plane for large values of the inverse temperature we construct an uncountable number of mutually singular gibbs states

the critical contact process dies out by producing a finite box criterion for survival in a slab we show that the critical contact process dies out and that in two and more dimensions the critical parameter value coincides with the limit of slab thresholds using the techniques developed in this paper one may obtain the complete convergence and shape theorems for contact processes in all dimensions and for all infection rates our results apply also to the discrete time analogue of the contact process viz oriented percolation

percolation critical exponents under the triangle condition for independent percolation models it is shown that if the diagrammatic triangle condition is satisfied then the critical exponents delta and hat beta exist and take their mean field values generalizing the criterion introduced in 1984 by aizenman and newman for the mean field value of gamma in nonoriented percolation the results apply to a broad class of nonoriented as well as oriented weakly homogeneous models in which the range of the connecting bonds need not be bounded for the nonoriented case the condition reduces to the finiteness at the critical point of nabla equiv sum x y tau 0 x tau x y tau y 0 with tau u nu the probability that the site u is connected to nu which was recently established by hara and slade for models with sufficiently spread out connections in d 6 dimensions our analysis proceeds through the derivation of complementary differential inequalities for the percolation order parameter m beta h whose value at h 0 yields the percolation density with beta parametrizing the bond or site occupation probabilities and with h h geq 0 a ghost field the conclusion is that under the triangle condition in the vicinity of the critical point beta c 0 m beta 0 approx beta beta c hat beta and m beta c h approx h 1 delta with hat beta 1 and delta 2

the random cluster model on a homogeneous tree the random cluster model on a homogeneous tree is defined and studied it is shown that for 1 q 2 the percolation probability in the maximal random cluster measure is continuous inp while forq 2 it has a discontinuity at the critical valuep p c q it is also shown that forq 2 there is nonuniqueness of random cluster measures for an entire interval of values ofp the latter result is in sharp contrast to what happens on the integer lattice z d

monotonicity of uniqueness for percolation on cayley graphs all infinite clusters are born simultaneously consider site or bond percolation with retention parameter p on an infinite cayley graph in response to questions raised by grimmett and newman 1990 and benjamini and schramm 1996 we show that the property of having almost surely a unique infinite open cluster is increasing in p moreover in the standard coupling of the percolation models for all parameters a s for all p2 p1 pc each infinite p2 cluster contains an infinite p1 cluster this yields an extension of alexander s 1995 simultaneous uniqueness theorem as a corollary we obtain that the probability v p that a given vertex v belongs to an infinite cluster is depends continuously on p throughout the supercritical phase p pc all our results extend to quasi transitive infinite graphs with a unimodular automorphism group

existence theorems for nonlinear boundary value problems for second order differential inclusions abstract in this paper we consider a nonlinear two point boundary value problem for second order differential inclusions using the leray schauder principle and its multivalued analog due to dugundji granas we prove existence theorems for convex and nonconvex problems our results are quite general and incorporate as special cases several classes of problems which are of interest in the literature

a homotopic deformation along p of a leray schauder degree result and existence for u p 2u ƒ t u 0 u 0 u t 0 p 1 soit f 0 1 r r une fonction continue et soit o p r r definie par o p s s p 2 s pour p fixe 1 on considere le probleme de dirichlet suivant o p u f t u 0 u 0 u t 0 ou t est un nombre reel positif

limit set of a weakly supercritical contact process on a homogeneous tree a conjecture of liggett 9 concerning the regime of weak survival for the contact process on a homogeneous tree is proved the conjecture is shown to imply that the hausdor dimension of the limit set of such a contact process is no larger than half the hausdor dimension of the space of ends of the tree the conjecture is also shown to imply that at the boundary between weak survival and strong survival the contact process survives only weakly a theorem previously proved by zhang 14 finally a stronger form of a theorem of hawkes and lyons concerning the hausdor dimension of a galton watson tree is proved

the ising model on diluted graphs and strong amenability say that a graph has persistent transition if the ising model on the graph can exhibit a phase transition nonuniqueness of gibbs measures in the presence of a nonzero external field we show that for nonamenable graphs for bernoulli percolation with p close to 1 all the infinite clusters have persistent transition on the other hand we show that for transitive amenable graphs the infinite clusters for any stationary percolation do not have persistent transition this extends a result of georgii for the cubic lattice a geometric consequence of this latter fact is that the infinite clusters are strongly amenable i e their anchored cheeger constant is 0 finally we show that the critical temperature for the ising model with no external field on the infinite clusters of bernoulli percolation with parameter p on an arbitrary bounded degree graph is a continuous function of p

nonlinear boundary value problems for second order differential inclusions abstract in this paper we study a class of nonlinear value boundary problems for second order differential inclusions with nonlinear perturbations which satisfy the generalized hartman condition weaker than that considered in some papers using techniques from multivalued analysis theory of monotone operators and fixed points we prove the existence of solutions in both convex and nonconvex cases our framework can be incorporated with dirichlet neumann and mixed boundary problems

growth profile and invariant measures for the weakly supercritical contact process on a homogeneous tree it is known that the contact process on a homogeneous tree of degree d 1 3 has a weak survival phase in which the infection survives with positive probability but nevertheless eventually vacates every finite subset of the tree it is shown in this paper that in the weak survival phase there exists a spherically symmetric invariant measure whose density decays exponentially at infinity thus confirming a conjecture of liggett the proof is based on a study of the relationships between various thermodynamic parameters and functions associated with the contact process initiated by a single infected site these include 1 the growth profile which determines the exponential rate of growth in space time on the event of survival 2 the exponential rate 3 of decay of the hitting probability function at infinity also studied by the author and 3 the exponential rate q of decay in time t of the probability that the initial infected site is infected at time t it is shown that 83 is a strictly increasing function of the infection rate a in the weak survival phase and that 3 1 fd at the upper critical point a2 demarcating the boundary between the weak and strong survival phases it is also shown that q 1 except at a2 where q 1

translation invariant equilibrium states of ferromagnetic abelian lattice systems the structure of the set of all translation invariant equilibrium states is determined for all temperatures for which the free energy is differentiable models with several phase transitions are discussed rigorously

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

existence of positive and of multiple solutions for nonlinear periodic problems abstract in this paper we consider a scalar periodic problem driven by the ordinary p laplacian differential operator and having a nonsmooth potential using a variational method based on nonsmooth critical point theory first we prove the existence of a strictly positive solution then by strengthening our hypotheses on the nonsmooth potential we prove the existence of a second periodic solution

periodic solutions for semi linear evolution inclusions in this paper we study the periodic problem for semi linear evolution inclusion using techniques from multivalued analysis and fixed point theorems we establish existence theorems under the one sided lipschitz condition first we prove existence theorems for nonconvex and convex problem second we look for extremal periodic solutions and prove the relaxation theorem

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

optimization of fourth order discrete approximation inclusions the paper concerns the necessary and sufficient conditions of optimality for cauchy problem of fourth order discrete pd and discrete approximate pda inclusions the main problem is the formulation of the fourth order adjoint discrete and discrete approximate inclusions dais and transversality conditions which are peculiar to problems including fourth order derivatives and approximate derivatives thus the necessary and sufficient conditions of optimality are proved incorporating the euler lagrange and hamiltonian forms of inclusions derivation of optimality conditions is based on the apparatus of locally adjoint mapping lam and equivalence of lams theorems moreover in the application of these results the fourth order linear optimal control problems with linear discrete and discrete approximate inclusions are considered an approach to obtain approximate numerical solutions of linear discrete inclusions is presented

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

a new proof that for the contact process on homogeneous trees local survival implies complete convergence we provide a new proof substantially simpler than zhang s original one that for the contact process on homogeneous trees local survival implies complete convergence

strongly nonlinear second order differential inclusions with generalized boundary conditions abstract in this paper we study second order differential inclusions in r n with a maximal monotone term and generalized boundary conditions the nonlinear differential operator need not be necessary homogeneous and incorporates as a special case the one dimensional p laplacian the generalized boundary conditions incorporate as special cases well known problems such as the dirichlet picard neumann and periodic problems as application to the proven results we obtain existence theorems for both convex and nonconvex problems when the maximal monotone term a is defined everywhere and when not case of variational inequalities

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

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

percolation on hyperbolic lattices the percolation transitions on hyperbolic lattices are investigated numerically using finite size scaling methods the existence of two distinct percolation thresholds is verified at the lower threshold an unbounded cluster appears and reaches from the middle to the boundary this transition is of the same type and has the same finite size scaling properties as the corresponding transition for the cayley tree at the upper threshold on the other hand a single unbounded cluster forms which overwhelms all the others and occupies a finite fraction of the volume as well as of the boundary connections the finite size scaling properties for this upper threshold are different from those of the cayley tree and two of the critical exponents are obtained the results suggest that the percolation transition for the hyperbolic lattices forms a universality class of its own

percolation in the hyperbolic plane the purpose of this paper is to study percolation in the hyperbolic plane and in transitive planar graphs that are quasi isometric to the hyperbolic plane

the ising model and percolation on trees and tree like graphs we calculate the exact temperature of phase transition for the ising model on an arbitrary infinite tree with arbitrary interaction strengths and no external field in the same setting we calculate the critical temperature for spin percolation the same problems are solved for the diluted models and for more general random interaction strengths in the case of no interaction we generalize to percolation on certain tree like graphs this last calculation supports a general conjecture on the coincidence of two critical probabilities in percolation theory

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

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

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

existence and relaxation results for nonlinear second order multivalued boundary value problems in rn abstract in this paper we study second order differential inclusions with nonlinear boundary conditions our formulation is general and incorporates as special cases well known problems such as the dirichlet picard neumann and periodic problems we prove existence theorems under various sets of hypotheses for both the convex and nonconvex problems also we show the existence of extremal solutions and that the extremal solutions are dense in the solutions of the convex problem for thew1 2 t r n norm strong relaxation theorem finally we examine the dirichlet problem when the multivalued right hand side does not depend on the derivative of x and satisfies a general growth hypothesis and a sign type condition for this problem we prove existence results and a relaxation theorem

critical probabilities for site and bond percolation models cc weak inequality p site p bond is strengthened to strict inequality for a cc broad category of graphs g including all the usual finite dimensional lattices in two and more dimensions the complementary inequality p site c bond 1 1 1 p is proved also where denotes the supremum of c the vertex degrees of g 0 introduction and results let g v e be an infinite connected graph our target in this paper is to study the relationship between site and bond percolation on g and particularly to prove inequalities between the two critical probabilities p site p bond of these models to date the only general cc inequality of this type appears to be the weak inequality p site p bond valid cc for all connected graphs g our principal purpose here is to extend the coupling arguments used in proving this weak inequality and to exploit recently developed methods for proving strict inequalities in order to obtain the inequality p site p bond for a certain broad category of graphs cc several difficulties arise in this program under two general headings first a method is required for utilizing the strict inequality methods of aizenman and grimmett 1991 in the nonstatic setting which occurs when studying stochastic couplings of site and bond percolation second there are graph theoretic complications in applying such techniques to general graphs we shall give two theorems about strict inequalities rather than one only our first such result is theorem 1 which applies specifically to hypercubic lattices in d 2 dimensions this will be proved using a relatively straightforward construction although this construction may in principle be ex tended with some combinatorial complications to certain other graphs on an ad hoc basis it does not seem to be easy to adapt it to a broad general class of graphs therefore we have separated out in theorem 2 our more general result the proof of which requires substantially deeper ideas than that of

stability of infinite clusters in supercritical percolation a recent theorem by haggstrom and peres concerning independent percolation is extended to all the quasi transitive graphs this theorem states that if 0 p1 p2 1 and percolation occurs at level p1 then every infinite cluster at level p2 contains some infinite cluster at level p1 consequences are the continuity of the percolation probability above the percolation threshold and the monotonicity of the uniqueness of the infinite cluster i e if at level p1 there is a unique infinite cluster then the same holds at level p2 these results are further generalized to graphs with a uniform percolation property the threshold for uniqueness of the infinite cluster is characterized in terms of connectivities between large balls

geometric analysis of φ4 fields and ising models parts i and ii we provide here the details of the proof announced in 1 that ind 4 dimensions the even 4 euclidean field theory with a lattice cut off is inevitably free in the continuum limit in the single phase regime the analysis is nonperturbative and is based on a representation of the field variables or spins in ising systems as source sink creation operators in a system of random currents which may be viewed as the mediators of correlations in this dual representation the onset of long range order is attributed to percolation in an ensemble of sourceless currents and the physical interaction in the 4 field and other aspects of the critical behavior in ising models are directly related to the intersection properties of long current clusters an insight into the criticality of the dimensiond 4 is derived from an analogy foreseen by k symanzik with the intersection properties of paths of brownian motion other results include the proof that in certain respect the critical behavior in ising models is in exact agreement with the mean field approximation in high dimensionsd 4 but not in the low dimensiond 2 for which we establish the universality of hyperscaling

existence and uniqueness results for some nonlinear boundary value problems u a s u b s0 dirichlet 2 u9 a s u9 b s 0 neumann 3 u a s u b and u9 a s u9 b periodic 4 w x here f r a r is an odd increasing homeomorphism k a b a r and w x 2 py2 f a b r a r equation 1 with f s s s s where p 1 and k t s n y 1 rt arises in the study of radial solutions for the p laplacian equation on an annular domain in n dimensions py2 n div u u q f x u u s 0 a x b x g r 5 w x and has been studied extensively see for instance 1 4 6 in this paper we shall establish existence results for 1 with each of the above boundary conditions 2 3 and 4 under various growth conditions on f in particular our results give the existence of radial solutions

explicit isoperimetric constants and phase transitions in the random cluster model the random cluster model is a dependent percolation model that has applications in the study of ising and potts models in this paper several new results are obtained for the random cluster model on nonamenable graphs with cluster parameter q geq 1 among these the main ones are the absence of percolation for the free random cluster measure at the critical value and examples of planar regular graphs with regular dual where p mathrm c mathrm free q p mathrm u mathrm wired q for q large enough the latter follows from considerations of isoperimetric constants and we give the first nontrivial explicit calculations of such constants such considerations are also used to prove nonrobust phase transition for the potts model on nonamenable regular graphs

on multiple phase transitions for branching markov chains we consider branching markov chains on a countable set we give a necessary and sufficient condition in terms of the transition kernel of the underlying markov chain to have two phase transitions we compute the critical values we apply this result to prove that asymmetric branching random walks onz have two phase transitions

the evaluation of service quality using generalized choquet integral in view of the compatibility between psychology and linguistic terms this study generalizes the standard choquet integral whose measurable evidence and fuzzy measures are real numbers the proposed generalization can deal with fuzzy number types of measurable evidence and fuzzy measures in comparison with the previous research on service quality and fuzzy sets this study integrates the three column format servqual into the generalized choquet integral a numerical example evaluating the overall service quality of e stores is provided to illustrate how the generalized choquet integral evaluates service quality in the three column format servqual this integral is also compared with the possibility necessity model when measuring service quality

multiple transition points for the contact process on the binary tree the contact process on z d is known to have only two fundamental types of behavior survival and extinction recently pemantle discovered that the phase structure for the contact process on a tree can be more complex there are three possible types of behavior strong survival weak survival and extinction he proved that all three occur on homogeneous trees in which each vertex has d 1 neighbors provided that d 3 but he left open the case d 2 since d 1 corresponds to z 1 in which weak survival does not occur d 2 is the boundary case in this paper we complete this picture by showing that weak survival does occur on the binary tree for appropriate parameter values in doing so we extend and develop techniques for obtaining upper and lower bounds for the critical values associated with strong and weak survival of the contact process on more general graphs

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

second order nonlinear multivalued boundary problems in hilbert spaces in this paper we consider second order differential inclusions in real hilbert space namely p t x t r t x t ax t f t x t a e on 0 t under the nonlinear boundary conditions using techniques from multivalued analysis and the theory of operators of monotone type we prove the existence of solutions for both the convex and nonconvex problems finally we present a special case of interest which fit into our framework illustrating the generality of our results

periodic problems for strongly nonlinear second order differential inclusions in this paper we study periodic problems for second order differential inclusions in rn with a maximal monotone term the nonlinear differential operator is not necessarily homogeneous does not obey any growth condition and incorporates as a special case the one dimensional p laplacian using techniques from multivalued analysis and the theory of operators of monotone type we prove the existence of solutions for both the convex and nonconvex problems when the maximal monotone term a is defined everywhere and when it is not defined everywhere case of variational inequalities

markov fields on branching planes we investigate ising models indexed by the sites of a branching plane mathbb t which is the product of a regular tree mathbb t and the line there are three parameter regimes corresponding to 1 a unique gibbs distribution 2 nonunique gibbs distributions with treelike structure the free boundary condition field is not a mixture of the plus and minus b c fields 3 nonunique gibbs distributions with planelike structure the free b c field is a mixture of the plus and minus b c fields

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

the ising model on hyperlattices the authors describe the construction of regular lattices in two dimensional hyperbolic space by means of the action of a discrete subgroups of su 1 1 they consider an ising model on such lattices and show how the thermodynamic limit can be handled they give high and low temperature expansions of the free energy magnetic susceptibility and magnetization and find that these quantities diverge at a critical temperature with mean field exponents beta 1 2 gamma 1 they also conjecture the long distance behaviour of correlation functions at the critical temperature

branching random walks and contact processes on homogeneous trees branching random walks and contact processes on the homogeneous tree in which each site has d 1 neighbors have three possible types of behavior for d 2 local survival local extinction with global survival and global extinction for branching random walks we show that if there is local extinction then the probability that an individual ever has a descendent at a site n units away from that individual s location is at most d n 2 while if there is global extinction this probability is at most d n next we consider the structure of the set of invariant measures with finite intensity for the system and see how this structure depends on whether or not there is local and or global survival these results suggest some problems and conjectures for contact processes on trees we prove some and leave others open in particular we prove that for some values of the infection parameter there are nontrivial invariant measures which have a density tending to zero in all directions and hence are different from those constructed by durrett and schinazi in a recent paper

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

existence of solutions for quasilinear second order differential inclusions with nonlinear boundary conditions abstract in this paper we consider a quasilinear second order differential inclusion with a convex valued multivalued term and nonlinear multivalued boundary conditions using the leray schauder fixed point theorem and techniques from multivalued analysis and from nonlinear analysis we prove the existence of a solution our formulation of the problem is general and includes as special cases the dirichlet the neumann the periodic problems as well as certain sturm liouville type problems