the magnetization energy scaling limit in high dimension in the single phase region including the critical point of a nearest neighbor ising ferromagnet with zero external field the block magnetization and energy within the infinite volume system are asymptotically for large block size independent gaussian variables when the dimensiond exceeds four for other models including ones with long range interactions a sufficient condition for such triviality of the scaling limit is finiteness of the bubble quantity

developments of some explicit formulas useful to describe elastic stress fields ahead of notches in plates abstract the paper deals with the evaluation of linear elastic stress fields in the neighbourhood of u and v shaped notches in plane plates the main aim is to improve the accuracy of an approximate solution already proposed in the literature by changing the polynomial arrangement of complex potential functions and properly adapting the boundary conditions in some special cases the solution matches williams westergaard irwin and creager paris equations universally used in the case of sharp corners cracks and blunt cracks respectively in the presence of notches with a tip radius different from zero and a large opening angle the equations obtained are compared with finite element results showing a very good agreement due to their reduced complexity such equations turn out to be particularly useful when applied to rounded v notches as well as welded like geometries under mixed loading conditions finally some limits of the solution when applied to the notch free edges are highlighted

upper bounds on the critical temperature for various ising models upper bounds are obtained for spin 1 systems in the case of only nearestneighbor interactions on for example the square lattice we obtain cj 0 3592 the method s strength is seen when considering systems with longer range interactions for example we obtain cj 0 360 compared to the previous best bound of c j 0 345 for the one dimensional lattice with 1 r2 interactions the method relies upon an identity between correlation functions and then the use of correlation inequalities to obtain the final bounds

stress analysis of quasi orthotropic elastic plane abstract the quasi orthotropic elastic plane in which the characteristic roots of the fundamental differential equation for the orthotropic elastic plane are doubled is investigated for the associated stress analysis a rational mapping function is used and a closed form solution is obtained therefore the stress analysis is rigorous for the rational mapping function the stress functions can be obtained without any integration as a demonstration of the stress analysis a half plane with an oblique edge crack is analyzed stress distributions and stress intensity factors are investigated the relationships between the quasi orthotropic elastic plane and isotropic plane with respect to the stress intensity factor are investigated it is also determined that the stress intensity factors for the quasi orthotropic elastic plane can be calculated from those of the isotropic elastic plane using a similar configuration and loading condition

splitting in large dimension and infrared estimates ii moment inequalities this is the continuation of notes written for the nato asi conference in il ciocco september 96 consisting of the analysis of the links between estimating the splitting between the two first eigenvalues for the schrodinger operator h and the proof of infrared estimates for quantities attached to gaussian type measures these notes were mainly reporting on the old contributions of dyson frohlich glimm jaffe lieb simon and spencer in the 1970s in connection with more recent contributions of pastur khoruzhenko barbulyak and kondrat ev which treat in general more sophisticated models here we concentrate on the simplest model related to field theory and extend the results of barbulyak and kondrat ev by mixing ideas coming from pastur and khozurenko related to the use of bogolyubov s inequality with classical inequalities due to ginibre lebowitz sokal and others or in the case when the temperature t is zero by applying rather elementary estimates on schrodinger operators in order to find

stress distribution at sharp and rounded v notches in quasi orthotropic plane abstract the eigensolution of the quasi orthotropic wedge problem is presented the degenerate orthotropic material of the wedge is characterized by doubled roots of the characteristic equation using the method of singular integral equations the distribution of the stress field in the quasi orthotropic plane weakened by semi infinite rounded v notch was obtained the edge of the notch was free of load and the asymptotic stress field defined by a stress intensity factor was set at the infinity on this basis the relationship between stress intensity factor at the sharp notch vertex and the normal stress at the vertex of corresponding rounded v notch with the same opening angle was established as it was shown in the example this relationship is asymptotic for finite notched bodies and it can be used to calculate a notch stress intensity factor based on a stress concentration results obtained for a notch rounded with small radius of curvature

a sinh transformation for evaluating two dimensional nearly singular boundary element integrals a new transformation technique is introduced for evaluating the two dimensional nearly singular integrals which arise in the solution of laplace s equation in three dimensions using the boundary element method when the source point is very close to the element of integration the integrals are evaluated using in a product fashion a transformation which has recently been used to evaluate one dimensional near singular integrals this sinh transformation method automatically takes into account the position of the projection of the source point onto the element and also the distance b between the source point and the element the method is straightforward to implement and when it is compared with a number of existing techniques for evaluating two dimensional near singular integrals it is found that the sinh method is superior to the existing methods considered both for potential integrals across the full range of b values considered 0 0 01 for smaller values of b the use of the lmethod is recommended for flux integrals copyright 2006 john wiley sons ltd

representations of o n spin models by self avoiding random walks we establish that correlation functions of classical lattice spin models can be represented by series expansions in terms of self avoiding random walks using this we get new upper bounds of critical temperatures of the o n symmetric classical heisenberg models

uniqueness and global markov property for euclidean fields the case of general polynomial interactions we give a general method for proving uniqueness and global markov property for euclidean quantum fields the method is based on uniform continuity of local specifications proved by using potential theoretical tools and exploitation of a suitable fkg order structure we apply this method to give a proof of uniqueness and global markov property for the gibbs states and to study extremality of gibbs states also in the case of non uniqueness in particular we prove extremality for 2 4 also in the case of non uniqueness and global markov property for weak coupling 2 4 which solves a long standing problem uniqueness and extremality holds also at any point of differentiability of the pressure with respect to the external magnetic field

hyper kahler metrics and harmonic superspace the most general unconstrained superfield action for self interactingn 2 matter hypermultiplets in analyticn 2 superspace is argued to produce a most generaln 2 hyper kahler model after eliminating an infinite set of auxiliary fields this suggests a new possibility of classifying hyper kahler metrics according to then 2 analytic superfield self interactions and provides an effective tool to compute these metrics explicitly as the simplest example the u 2 invariant quartic self coupling of a singleq hypermultiplet is analyzed and is shown to yield the familiar taub nut metric to see the geometric pattern directly in terms ofn 2 superfields we introduce a new on shell representation ofq hypermultiplets inn 2 harmonic superspace similar to the description ofn 2 gauge theories for the u 2 example this formulation is checked to coincide with that by sierra and townsend

hyperkahler manifolds and nonlinear supermultiplets we present a new construction of hyperkahler metrics that derives from the 3 dimensionaln 4 nonlinear supermultiplet further we give a detailed description of the nonlinear multiplet inn 2 and 4 superspace

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

self avoiding random walks on the hexagonal lattice we use the algorithm recently introduce by a berretti and a d sokal to compute numerically the critical exponents for the self avoiding random walk on the hexagonal lattice we find 1 3509 0 0057 0 0023v 0 7580 0 0049 0 0046 0 519 0 082 0 077 where the first error is the systematic one due to corrections to scaling and the second is the statistical error for the effective coordination number we find 1 84779 0 00006 0 0017 the results support the nienhuis conjecture 43 32 and provide a rough numerical check of the hyperscaling relationdv 2 an additional analysis taking the nienhuis value of 2 21 2 1 2 for granted gives 1 3459 0 0040 0 0008

correlations equalities and some upper bounds for the critical temperature for spin one systems abstract starting from correlation identities for the blume capel spin 1 systems and using correlation inequalities we obtain rigorous upper bounds for the critical temperature the obtained results improve over effective field type results

upper bound on the critical temperature in the 3d ising model the upper bound on the critical temperature in the ising model recently derived from callen s correlation equalities by sa barreto and o carroll 1983 and improved by monroe 1984 is further improved by a more economical use of messager and miracle sole inequalities starting from a different callen correlation equality

markov processes as a tool in field theory abstract in a recent paper brydges frohlich and spencer have successfully applied markov chains to classical spin systems using probabilistic methods in a more systematic way we improve some of their results the methods can be extended to fields with continuous parameter such as p fields

skeleton inequalities and mean field properties for lattice spin systems we present a proof of skeleton inequalities for ferromagnetic lattice spin systems with potential v 2 a 2 2 mn 2 2n 2n 2n a real 2n 0 generalizing the brydges frohlich sokal and bovier felder methods as an application of the inequalities we prove that for sufficiently soft systems in d 4 dimensions critical exponents and 4 take their mean field values i e 1 0 and 4 3 2

a phase cell cluster expansion for a hierarchical o 3 4 model the formalism developed in a previous paper is applied to yield a phase cell cluster expansion for a hierarchical b model the field is expanded into modes with specific renormatization group scaling properties the present cluster expansion for a vacuum expectation value is formally the natural factorization of each term in the perturbation expansion into the contribution of modes connected to the variables in the expectation via interactions and that of the complementary set the expectation value is thus realized as a sum of contributions due to finite subsets of the modes we emphasize the following additional features 1 partitions of unity are not used 2 there are essentially no cut offs 3 the expansion is developed directly without an initial need to prove an ultraviolet stability bound the most difficult part of the traditional approach our main interest in the present phase cell cluster expansion is founded in the belief that it may be the right vehicle for proving the existence of a non trivial four dimensional field theory

borel summability of the 1 n expansion for then vector o n non linear σ models we construct an analytic interpolation in 1 n for then vector o n non linear models withn component fields on a lattice this interpolation valid at sufficiently high temperatures extends over a large domain in the complex plane containing the half plane re 1 n 0 we use this result to show that the 1 n expansion of the free energy density and of the correlation functions is borel summable in the thermodynamic limit and at high temperature

a unified solution approach for a large variety of antiplane shear and torsion notch problems theory and examples abstract in this work a unified solution approach is proposed for the analytical evaluation of the stress fields close to notches under antiplane shear and torsion loadings which allows a large variety of notch problems to be tackled the method is based on the complex potential approach for antiplane elasticity combined with the use of proper conformal mappings in particular it is shown that a well defined analytical link does exist between the complex potential to be used to determine stresses and the first derivative of the conformal mapping used to mathematically describe the notch profile this makes some methodologies such as schwarz christoffel transformation which allows describing any polygonal domain automatically very attractive for the direct solution of notch problems a bulk of solutions are provided to support this finding from cracks and pointed notches to radiused notches in addition the accuracy of each proposed solution is discussed in detail taking advantage of a bulk of results from fe analyses

mean field bounds and correlation inequalities i prove a new correlation inequality for a class ofn component classical ferromagnets 1 n 4 this inequality implies that the correlation functions decay exponentially and the spontaneous magnetization is zero above the mean field critical temperature

rigorous studies of critical behavior the task addressed in the talk is the analysis of the critical behavior in statistical mechanical systems and related problems in the theory of random fields while it would have been very useful to have mathematical means of approaching this subject at the level of generality presented here in professor dobrushin s analysis of the high temperature phases we find that the richness of the behavior exhibited at low temperatures calls for more restricted treatments of special models nevertheless it was found that the somewhat delicate issues of the phase structure and the critical behavior may be approached by methods which apply to a number of systems in particular the picture offered by random walk representations led to very useful insights for the study of ferromagnetic systems 4 d field theory and percolation models it is hoped that random surface expansions might play a somewhat similar role for gauge fields and some models of spin glasses

higgs 2 3 quantum fields in a finite volume we consider a euclidean model of interacting scalar and vector fields in two and three dimensions and prove a lower bound for vacuum energy in a lattice approximation the bound is independent of a lattice spacing it is proved with the help of renormalization transformations in wilson kadanoff form it extends in principal also to generating functional for schwinger functions

n 2 super yang mills theory in projective superspace we constructn 2 yang mills theory in projective superspace by exploiting the analogy to ward s twistor construction of self dual yang mills fields

upper bounds on the critical temperature for the two dimensional blume emery griffiths model we obtain rigorous upper bounds for the critical temperature associated with second order phase transitions of the two dimensional spin 1 beg model for real values ofk andd coupling constants and forj 0 we use some correlation equalities and inequalities to show the exponential decay of the two point function characterizing the disordered phase

correlation inequalities for two component hypercubic ϕ 4 models a collection of new and already known correlation inequalities is found for a family of two component hypercubic var phi sub 4 models using techniques of duplicated variables rotated correlation inequalities and random walk representation among the interesting new inequalities are rotated very special dunlop newman inequality greater than or equal to 0 and anti lebowitz inequality u sub 4 sup 1111 greater than or equal to 0

a sinh transformation for evaluating nearly singular boundary element integrals an implementation of the boundary element method requires the accurate evaluation of many integrals when the source point is far from the boundary element under consideration a straightforward application of gaussian quadrature suffices to evaluate such integrals when the source point is on the element the integrand becomes singular and accurate evaluation can be obtained using the same gaussian points transformed under a polynomial transformation which has zero jacobian at the singular point a class of integrals which lies between these two extremes is that of nearly singular integrals here the source point is close to but not on the element and the integrand remains finite at all points however instead of remaining flat the integrand develops a sharp peak as the source point moves closer to the element thus rendering accurate evaluation of the integral difficult this paper presents a transformation based on the sinh function which automatically takes into account the position of the projection of the source point onto the element which we call the nearly singular point and the distance from the source point to the element the transformation again clusters the points towards the nearly singular point but does not have a zero jacobian implementation of the transformation is straightforward and could easily be included in existing boundary element method software it is shown that for the two dimensional boundary element method several orders of magnitude improvement in relative error can be obtained using this transformation compared to a conventional implementation of gaussian quadrature asymptotic estimates for the truncation errors are also quoted copyright 2004 john wiley sons ltd

stresses in the region of rounded corners abstract the mode i state of stress around a crack front i e in a wedge of angle 2 is well defined by the stress intensity factor k i the stress y k 1 2 r becomes infinite at the vertex r 0 it is known that if the sharp crack is rounded by a radius small with respect to crack length and other dimensions the maximum value for the stress y only becomes y k 1 by numerical solution of integral equations corresponding results are given for wedges of arbitrary angles

critical point dominance in one dimension the renormalized dimensionless 4 point coupling constant of scalar one dimensional field theories is maximized uniquely by the critical point theories obtainable as the scaling limit of 4 models the renormalized coupling constant of certain scalar one dimensional lattice field theories is maximized uniquely for fixed correlation length by the corresponding spin 1 2 model

walk sums and belief propagation in gaussian graphical models we present a new framework based on walks in a graph for analysis and inference in gaussian graphical models the key idea is to decompose the correlation between each pair of variables as a sum over all walks between those variables in the graph the weight of each walk is given by a product of edgewise partial correlation coefficients this representation holds for a large class of gaussian graphical models which we call walk summable we give a precise characterization of this class of models and relate it to other classes including diagonally dominant attractive non frustrated and pairwise normalizable we provide a walk sum interpretation of gaussian belief propagation in trees and of the approximate method of loopy belief propagation in graphs with cycles the walk sum perspective leads to a better understanding of gaussian belief propagation and to stronger results for its convergence in loopy graphs

the random walk representation of classical spin systems and correlation inequalities ii the skeleton inequalities on utilise la representation marche aleatoire pour prouver les premieres d une nouvelle famille d inegalites de correlation pour des modeles reticules ferromagnetiques 4 ces inegalites etablissent que les sommes partielles finies du developpement en perturbation resomme du propagateur pour la fonction a 4 points forment un ensemble alterne de limites superieures et inferieures rigoureuses pour la fonction a 4 points exactes

skeleton inequalities and the asymptotic nature of perturbation theory for φ4 theories in two and three dimensions we use the polymer representation of 4 quantum field theories to prove an infinite family of correlation inequalities called skeleton inequalities for the 2n point green s functions as an application we show that they imply that feynman perturbation theory is asymptotic in less than four dimensions

the first passage sets of the 2d gaussian free field convergence and isomorphisms in a previous article we introduced the first passage set fps of constant level a of the two dimensional continuum gaussian free field gff on finitely connected domains informally it is the set of points in the domain that can be connected to the boundary by a path along which the gff is greater than or equal to a this description can be taken as a definition of the fps for the metric graph gff and in the current article we prove that the metric graph fps converges towards the continuum fps in the hausdorff distance we also draw numerous consequences in particular we obtain a relatively simple proof of the fact that certain natural interfaces of the metric graph gff converge to hbox sle 4 level lines these results improve our understanding of the continuum gff by strengthening its relationship with the critical brownian loop soup indeed a new construction of the fps using clusters of brownian loops and excursions helps to strengthen the known gff isomorphism theorems and allows us to use brownian loop soup techniques to prove technical results on the geometry of the gff we also obtain a new representation of brownian loop soup clusters and as a consequence we prove that the clusters of a critical brownian loop soup admit a non trivial minkowski content in the gauge r mapsto log r 1 2 r 2

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

singularities interfaces and cracks in dissimilar anisotropic media for a non pathological bimaterial in which an interface crack displays no oscillatory behaviour it is observed that apart possibly from the stress intensity factors the structure of the near tip field in each of the two blocks is independent of the elastic moduli of the other block collinear interface cracks are analysed under this non oscillatory condition and a simple rule is formulated that allows one to construct the complete solutions from mode iii solutions in an isotropic homogeneous medium the general interfacial crack tip field is found to consist of a two dimensional oscillatory singularity and a one dimensional square root singularity a complex and a real stress intensity factors are proposed to scale the two singularities respectively owing to anisotropy a peculiar fact is that the complex stress intensity factor scaling the oscillatory fields however defined does not recover the classical stress intensity factors as the bimaterial degenerates to be non pathological collinear crack problems are also formulated in this context and a strikingly simple mathematical structure is identified interactive solutions for singularity interface and singularity interface crack are obtained the general results are specialized to decoupled antiplane and in plane deformations for this important case it is found that if a material pair is non pathological for one set of relative orientations of the interface and the two solids it is non pathological for any set of orientations for bonded orthotropic materials an intuitive choice of the principal measures of elastic anisotropy and dissimilarity is rationalized a complex variable representation is presented for a class of degenerate orthotropic materials throughout the paper the equivalence of the lekhnitskii and stroh formalisms is emphasized the article concludes with a formal statement of interfacial fracture mechanics for anisotropic solids

new hyperkahler metrics and new supermultiplets we present new constructions of hyperkahler metrics along with the three new classes ofn 4 supermultiplets that they derive from further we provide a general setting for understanding the constructions and give a detailed description of the multiplets inn 2 andn 4 superspace

lectures on nonlinear sigma models in projective superspace n 2 supersymmetry in four space time dimensions is intimately re lated to hyperkahler and quaternionic kahler geometries on one hand the target spaces for rigid supersymmetric sigma models are necessarily hy perkahler manifolds on the other hand when coupled to n 2 supergrav ity the sigma model target spaces must be quaternionic kahler it is known that such manifolds of restricted holonomy are dicult to generate explic itly projective superspace is a eld theoretic approach to construct general n 2 supersymmetric nonlinear sigma models and hence to generate new hyperkahler and quaternionic kahler metrics intended for a mixed audience consisting of both physicists and mathematicians these lectures provide a pedagogical introduction to the projective superspace approach

a monte carlo analysis of self avoiding walks in three dimensions the authors present estimates for critical exponents of self avoiding walks on a cubic lattice they treat a grand canonical ensemble of walks with free ends in a monte carlo approach and make use of real space renormalisation ideas their estimate for v is 0 59 or 0 01 the estimate for gamma does not share the same degree of accuracy however they are able to pinpoint the source of this discrepancy in addition they define a quantity chi l beta the probability that a walk starting at the origin will end outside or at the border of a cube of side l this quantity turns out to be quite suitable for a real space renormalisation group analysis

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

hyperkahler metrics and supersymmetry we describe two constructions of hyperkahler manifolds one based on a legendre transform and one on a sympletic quotient these constructions arose in the context of supersymmetric nonlinear models but can be described entirely geometrically in this general setting we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience

on the determination of higher order terms of singular elastic stress fields near corners in two dimensional elasticity stresses at reentrant corners exhibit singular behavior the stress field is of the form sigma sigma rm k iota r lambda 2 1 f i theta lambda i where r are polar coordinates centered at the tip of the corner andf i i are smooth functions for practical use of this series the eigenvalues i which are generally complex numbers are required in order of ascending real part the problem then is to find the roots of a transcendental equation eigenequation in the complex plane and arranged in order of ascending real part a theorem is proved on the number location and nature of the roots of this equation with the real part in fixed intervals of length excellent initial estimates of the real part of the complex roots become available and so are bounds within which single real roots exist this enables the determination of any number of roots in ascending order of real part the critical angles at which the eigenvalues change nature are also determined it is shown that for certain cases and for the symmetric mode of deformation the eigenvalue 1 does not represent a rigid body rotation therefore it has to be included in the series representation of the stresses the coefficientsk i can be determined by recently developed extraction techniques thus allowing complete determination of the elastic field and enabling its correlation with experimental data on brittle fracture crack initiation plastic zone estimation etc

upper bounds on tc for classical ising models using correlation inequalities following closely the work of sa barreto and o carroll 1983 the author improves the upper bounds they derive for the critical temperature of classical ising ferromagnetics the improvement is achieved by a more critical use of griffith s 1967 inequalities at one point along with a completely new use of messager and miracle sole 177 inequalities at another stage of the proof

random fields associated with multiple points of the brownian motion abstract let xt be the brownian motion in r d the random set t1 tn z xtl xtn z in r d n is empty a s except in the following cases a n 1 d 1 2 b d 2 n 2 3 c d 3 n 2 in each of these cases a family of random measures m concentrated on is constructed takes values in a certain class of measures on r d measures m characterize the time space location of self intersections for brownian paths if n d 1 then m dt dz dz nz dt where n2 is the local time at z in the case n 2 the set can be identified with the set of brownian loops the measure m explodes on the diagonal t1 t2 and to study small loops a random distribution which regularizes m is constructed

comments on n 2 supersymmetric σ models in projective superspace for the most general off shell supersymmetric model in projective superspace we elaborate on its formulation in terms of chiral superfields a universal model independent expression is obtained for the holomorphic symplectic two form which determines the second supersymmetry transformation this two form is associated with the two complex structures of the hyperk hler target space which are complimentary to the one used to realize the target space as a k hler manifold

pinning of an interface by a weak potential we prove that in a two dimensional gaussian sos model with a small attractive potential the height of the interface remains bounded no matter how small the potential is this is in sharp contrast with the free situation in which the interface height diverges logarithmically in the thermodynamic limit

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