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

dimensional reduction for generalized continuum polymers the brydges imbrie dimensional reduction formula relates the pressure of a d dimensional gas of hard spheres to a model of d 2 dimensional branched polymers brydges and imbrie s proof was non constructive and relied on a supersymmetric localization lemma the main result of this article is a constructive proof of a more general dimensional reduction formula that contains the brydges imbrie formula as a special case central to the proof are invariance lemmas which were first introduced by kenyon and winkler for branched polymers the new dimensional reduction formulas rely on invariance lemmas for central hyperplane arrangements that are due to meszaros and postnikov several applications are presented notably dimensional reduction formulas for i non spherical bodies and ii for corrections to the pressure due to symmetry effects

a geometric interpretation of the roughening transition in ising models progress in the area of the ising model roughening transition has previously been limited by the lack of a good definition for the interface separating the pure phases in the present work a graphical definition is introduced and it is shown that roughening occurs precisely when this interface fluctuates to infinity

correlation inequalities for the truncated two point function of an ising ferromagnet we establish the following new correlation inequalities for the truncated twopoint function of an ising ferromagnet in a positive external field j l t j k t k l t and j l t k k j k t k l wherek is any set of sites which separatesj froml the inequalities are also valid for the pure phases with zero magnetic field at all temperatures above the critical temperature they reduce to known inequalities of griffiths and simon respectively

critical exponents for the self avoiding random walk in three dimensions we compute by direct monte carlo simulation the main critical exponents 4 andv and the effective coordination number for the self avoiding random walk in three dimensions on a cubic lattice we find both hyperscaling relationsdv 2 anddv 2 4 0 satisfied ind 3

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

on the upper critical dimension of lattice trees and lattice animals we give a rigorous proof of mean field critical behavior for the susceptibility 1 2 and the correlation length v 1 4 for models of lattice trees and lattice animals in two cases i for the usual model with trees or animals constructed from nearest neighbor bonds in sufficiently high dimensions and ii for a class of spread out or long range models in which trees and animals are constructed from bonds of various lengths above eight dimensions this provides further evidence that for these models the upper critical dimension is equal to eight the proof involves obtaining an infrared bound and showing that a certain square diagram is finite at the critical point and uses an expansion related to the lace expansion for the self avoiding walk

large scale behavior of three dimensional continuous phase coexistence models we study a class of three dimensional continuous phase coexistence models and show that under different symmetry assumptions on the potential the large scale behavior of such models near a bifurcation point is described by the dynamical urn x wiley 00103640 media cpa21738 cpa21738 math 0001 models for p 2 3 4 this result is specific to space dimension 3 and does not hold in dimension 2

lattice permutations and poisson dirichlet distribution of cycle lengths we study random spatial permutations on 3 where each jump x x is penalized by a factor mathrm e t x pi x 2 the system is known to exhibit a phase transition for low enough t where macroscopic cycles appear we observe that the lengths of such cycles are distributed according to poisson dirichlet this can be explained heuristically using a stochastic coagulation fragmentation process for long cycles which is supported by numerical data

on klauder s pseudo free oscillator studies the quantum oscillator on the half line defined by the hamiltonian h 1 2 delta 2 delta x2 1 2x2 together with vanishing boundary conditions at 0 the authors derives and discusses a new operator equation of motion of the form x x a for this system this equation is used to verify j r klauder s conjecture 1979 that x satisfy x x x 0 it is also used to obtain systematic results on the set of n point functions omega mod x t1 x t2 x tn mod omega it is suggested that these methods and results will act as a source of conjectures about the analogous pseudo free field theories postulated by klauder in his program for handling non renormalisable field theories

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

a note on the critical dimension for weakly self avoiding walks we consider a weakly self avoiding t step random walk on zdwith possible nonnearest neighbor jumps we prove that for d 3 the scaling limit of the end point is cauchy distributed as t

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

monte carlo test of a hyperscaling relation for the two dimensional self avoiding walk the authors simulated self avoiding walks on the square lattice with fixed endpoints by means of a dynamic monte carlo algorithm from these data they obtain an evaluation of the effective coordination number mu and the critical exponents alpha and v they can therefore test the hyperscaling relation 2 alpha dv with a careful estimate of systematic and statistical errors

logarithmic correction for the susceptibility of the 4 dimensional weakly self avoiding walk a renormalisation group analysis we prove that the susceptibility of the continuous time weakly self avoiding walk on mathbb z d in the critical dimension d 4 has a logarithmic correction to mean field scaling behaviour as the critical point is approached with exponent frac 1 4 for the logarithm the susceptibility has been well understood previously for dimensions d 5 using the lace expansion but the lace expansion does not apply when d 4 the proof begins by rewriting the walk two point function as the two point function of a supersymmetric field theory the field theory is then analysed via a rigorous renormalisation group method developed in a companion series of papers by providing a setting where the methods of the companion papers are applied together the proof also serves as an example of how to assemble the various ingredients of the general renormalisation group method in a coordinated manner

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

rigorous analysis of discontinuous phase transitions via mean field bounds we consider a variety of nearest neighbor spin models defined on the d dimensional hypercubic lattice z d our essential assumption is that these models satisfy the condition of reflection positivity we prove that whenever the associated mean field theory predicts a discontinuous transition the actual model also undergoes a discontinuous transition which occurs near the mean field transition temperature provided the dimension is sufficiently large or the first order transition in the mean field model is sufficiently strong as an application of our general theory we show that for d sufficiently large the 3 state potts ferromagnet on z d undergoes a first order phase transition as the temperature varies similar results are established for all q state potts models with q 3 the r component cubic models with r 4 and the o n nematic liquid crystal models with n 3

intersections of random walks a direct renormalization approach various intersection probabilities of independent random walks ind dimensions are calculated analytically by a direct renormalization method adapted from polymer physics this heuristic approach based on edwards continuum model leads to a straightforward derivation and also to refinements of lawler s results for the simultaneous intersections of two walks in 4 or three walks in 3 these results are generalized top walks in d d frac 2p p 1 p 2 ford 4 an infinite set of universal critical exponents l l 1 are derived they govern the asymptotic probability z l sim s sigma l thatl star walks in d with a common origin do not intersect before times the l s are calculated up to ordero e2 whered 4 e this information is used to calculate the probability z g that a set of independent random walks in d or d d 4 respectivelyd 3 form a given topological networks g of multiple intersection points in the absence of any other double point respectively triple point this is generalized to a network in d leqq frac 2p p 1 dimension with exclusion ofp tuple points the method is quite general and can be used to calculate any critical intersection probability and provides the probabilist with a large variety of exact results yet to be proven rigorously

loop erased random walk on a torus in dimensions 4 and above we show that the statistics of loop erased random walks above the upper critical dimension 4 are different between the torus and the full space the typical length of the path connecting a pair of sites at distance l which scales as l2 in the full space changes under the periodic boundary conditions to ld 2 the results are precise for dimensions 5 for the dimension d 4 we prove an upper bound conjecturally sharp up to subpolyonmial factors

mean field critical behaviour for correlation length for percolation in high dimensions extending the method of 27 we prove that the corrlation length of independent bond percolation models exhibits mean field type critical behaviour i e p p c p 1 2 asp p c in two situations i for nearest neighbour independent bond percolation models on ad dimensional hypercubic lattice d withd sufficiently large and ii for a class of spread out independent bond percolation models which are believed to belong to the same universality class as the nearest neighbour model in more than six dimensions the proof is based on and extends a method developed in 27 where it was used to prove the triangle condition and hence mean field behaviour of the critical exponents and 2 for the above two cases

radiation effects for the blasius and sakiadis flows with a convective surface boundary condition abstract this study aims to analyze the effects of thermal radiation on the laminar boundary layer about a flat plate in a uniform stream of fluid blasius flow and about a moving plate in a quiescent ambient fluid sakiadis flow both under a convective surface boundary condition the resulting similarity energy equation is solved numerically and the variations of dimensionless surface temperature and fluid solid interface characteristics for different values of prandtl number pr radiation parameter nr and parameter a which characterizes our convection processes are graphed and tabulated quite different and interesting behaviours were encountered for a blasius flow compared with a sakiadis flow a comparison with previously published results on special case of the problem shows excellent agreement

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

non gibbsian limit for large block majority spin transformations we generalize a result of lebowitz and maes that projections of massless gaussian measures onto ising spin configurations are non gibbs measures this result provides the first evidence for the existence of singularities in majority spin transformations of critical models indeed under the assumption of the folk theorem that an average block spin transformation applied to a critical ising model in 5 or more dimensions converges to a gaussian fixed point we show that the limit of a sequence of majority spin transformations with increasing block size applied to a critical ising model is a measure that is not of gibbsian type

discrete representation of straight lines if a continuous straight line segment is digitized on a regular grid obviously a loss of information occurs as a result the discrete representation obtained e g a chaincode string can be coded more conveniently than the continuous line segment but measurements of properties such as line length performed on the representation have an intrinsic inaccuracy due to the digitization process in this paper two fundamental properties of the quantization of straight line segments are treated 1 it is proved that every straight chaincode string can be represented by a set of four unique integer parameters definitions of these parameters are given 2 a mathematical expression is derived for the set of all continuous line segments which could have generated a given chaincode string the relation with the chord property is briefly discussed

fluid flow over a nonlinearly stretching sheet numerical solutions are obtained for a class of nonlinear third order differential equations arising in fluid flows over a nonlinearly stretching sheet using a similarity transformation which is different from that of the linearly stretching sheet problem furthermore using the schauder theory existence of a solution of the third order differential equation over a large but finite interval is established moreover the analytical solutions are obtained and are compared with the corresponding numerical solutions the results obtained for the flow characteristics reveal many interesting behaviors that warrant further study of the effects of nonlinear stretching on the flow characteristics

quantum field theory models on fractal spacetime continuing the analysis of a previous paper the present work applies rigorous renormalization group methods to the hierarchical models to establish the existence of field theories with non gaussian ultraviolet renormalization group fixed points in 4 e dimensions

a new solution branch for the blasius equation a shrinking sheet problem in this work a similarity equation of the momentum boundary layer is studied for a moving flat plate with mass transfer in a stationary fluid the solution is applicable to the practical problem of a shrinking sheet with a constant sheet velocity theoretical estimation of the solution domain is obtained it is shown that the solution only exists with mass suction at the wall surface the equation with the associated boundary conditions is solved using numerical techniques greatly different from the continuously stretching surface problem and the blasius problem with a free stream quite complicated behavior is observed in the results it is seen that there are three different solution zones divided by two critical mass transfer parameters f 0 1 1 7028 and f 0 2 1 7324 when f 0f 0 2 there is a terminating point for the solution domain and the terminating point corresponds to a special algebraically decaying solution for the current problem the current results provide a new solution branch of the blasius equation which is greatly different from the previous study and provide more insight into the understanding of the blasius equation

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

proprietes d intersection des marches aleatoires i convergence vers le temps local d intersection we study intersection properties of multi dimensional random walks letx andy be two independent random walks with values in d d 3 satisfying suitable moment assumptions and letin denote the number of common points to the paths ofx andy up to timen the sequence in suitably normalized is shown to converge in distribution towards the intersection local time of two independent brownian motions results are applied to the proof of a central limit theorem for the range of a two dimensional recurrent random walk thus answering a question raised by n c jain and w e pruitt

renormalization group fixed points in the local potential approximation ford 3 in the local potential approximation renormalization group equations reduce to a semilinear parabolic partial differential equation felder 8 has derived this equation and has constructed a family of non trivial fixed pointsu 2n n 2 3 4 which have the form ofn well potentials and exist in the ranges of dimensions 2 d 2 2 n 1 in this paper we show that ifd 4 then these non trivial fixed points disappear and if 3 d 4 then we have only theu 4 fixed point

quantum interaction φ44 the construction of quantum field defined as a bilinear form we construct the solution t x of the quantum wave equation m2 3 0 as a bilinear form which can be expanded over wick polynomials of the free in field and where 3 t x is defined as the normal ordered product with respect to the free in field the constructed solution is correctly defined as a bilinear form on d d where d is a dense linear subspace in the fock space of the free in field on d d the diagonal of the wick symbol of this bilinear form satisfies the nonlinear classical wave equation

random walks polymers percolation and quantum gravity in two dimensions we consider l planar random walks or brownian motions of large length t starting at neighboring points and the probability pl t t l that their paths do not intersect by a 2d quantum gravity method i e a non linear map onto a random riemann surface the former conjecture that l 124 4l2 1 is established this also applies to the half plane where l l3 1 2l as well as to non intersection exponents of unions of paths mandelbrot s conjecture for the hausdorff dimension dh 43 of the frontier of a brownian path follows from dh 2 32 we then consider in two dimensions the most general star shaped copolymer mixing random rw or self avoiding walks saw with specific mutual avoidance interactions thereof its exact conformal scaling dimensions in the plane are derived the harmonic measures or electrostatic potential or diffusion field near a rw or a saw or near a critical percolation cluster are also considered their moments exhibit a multifractal spectrum the generalized dimensions d n as well as the function are derived and are shown to be all identical these are examples of exact conformal multifractality

gaussian bounds for correlations in lattice spin systems in quite generaln component ferromagnetic spin systems it is proved that an arbitrary correlation function is bounded by the corresponding correlation function of a gaussian model the bound is useful for the analysis of high temperature behavior of the system similar bounds for truncated correlation functions are also obtained for a class of single component spin systems

mean field bound and ghs inequality a new proof of the mean field bounds for magnetizations is presented it applies to any single component spin system which allows ghs inequality and to ann vector model forn 3 and to ann solid sphere model for all values ofn provided that the interactions are ferromagnetic and translation invariant

fluctuation susceptibility relations for classical spin systems we prove identities between integrated ursell functions and derivatives of the pressure in the thermodynamic limit for multicomponent classical spin systems which obey the lee yang theorem and some form of gaussian domination when the susceptibility is finite t tc following refs 3 and 4 we view the moment generating function of the magnetization as the inverse of an infinitely divisible characteristic function fluctuation susceptibility relations of all orders then follow by bounding the corresponding cumulants taken in zero external field high order cumulants are bounded in terms of the susceptibility using gaussian and simon s inequalities for short range interactions

viscous flow over a nonlinearly stretching sheet an analysis is carried out to study the flow and heat transfer characteristics in a viscous fluid over a nonlinearly stretching sheet the equations of conservation of momentum mass and energy which govern the flow and heat transfer are partially decoupled using a similarity transformation which is different from that of the linearly stretching sheet problem however the results for the linearly stretching sheet problem can be obtained as a special case the basic nonlinear differential equation for the velocity field f and the differential equation with variable coefficient for the temperature field q are solved numerically by using a fourth order runge kutta integration scheme the numerical results thus obtained for the flow and heat transfer characteristics are presented by a table and graphs furthermore it is shown that the heat flow is always from the stretching sheet to the fluid

nontrivial path integrals for nonrenormalizable fields multicomponent ultralocal models a nontrivial lattice space path integral formulation of nonrenormalizable multicomponent ultralocal models is constructed from the nonperturbative operator solutions presented in a recent paper the indefinite nonclassical singular potential required for the nontriviality has different effects on distributions compared to the single component case however the essential property of reweighting the distribution at the origin is similar the appearance of additional nonclassical singular potentials suggests that we cannot always place the classical lagrangian or classical hamiltonian directly into the path integral formulation or in other words a straightforward canonical quantization of fields with infinite degrees of freedom does not always apply

operator analysis of nonrenormalizable multicomponent ultralocal field models a nontrivial nonperturbative quantum field theory of o n invariant multicomponent nonrenormalizable ultralocal models is developed along lines presented earlier for single component models the additional nonclassical repulsive potential that is always present in the solution of the single component case becomes indefinite and may even vanish in the multicomponent case the disappearance of the nonclassical and singular potential does not mean a return to a conventional quantum field theory and the operator solution of multicomponent ultralocal fields remains noncanonical nontrivial i e non gaussian results hold for any number n of components and suitable nontrivial behavior persists even in the infinite component n case as well

the analysis of the widom rowlinson model by stochastic geometric methods we study the continuum widom rowlinson model of interpenetrating spheres using a new geometric representation for this system we provide a simple percolation based proof of the phase transition we also use this representation to formulate the problem and prove the existence of an interfacial tension between coexisting phases finally we ascribe geometric i e probabilistic significance to the correlation functions which allows us to prove the existence of a sharp correlation length in the single phase regime

a pfaffian formula for monomer dimer partition functions we consider the monomer dimer partition function on arbitrary finite planar graphs and arbitrary monomer and dimer weights with the restriction that the only non zero monomer weights are those on the boundary we prove a pfaffian formula for the corresponding partition function as a consequence of this result multipoint boundary monomer correlation functions at close packing are shown to satisfy fermionic statistics our proof is based on the celebrated kasteleyn theorem combined with a theorem on pfaffians proved by one of the authors and a careful labeling and directing procedure of the vertices and edges of the graph

on recursive o n partitioning of a digitized curve into digital straight segments a simple online algorithm for partitioning of a digital curve into digital straight line segments of maximal length is given the algorithm requires o n time and o 1 space and is therefore optimal efficient representations of the digital segments are obtained as byproducts the algorithm also solves a number theoretical problem concerning nonhomogeneous spectra of numbers

computational geometry an introduction from the reviews this book offers a coherent treatment at the graduate textbook level of the field that has come to be known in the last decade or so as computational geometry the book is well organized and lucidly written a timely contribution by two founders of the field it clearly demonstrates that computational geometry in the plane is now a fairly well understood branch of computer science and mathematics it also points the way to the solution of the more challenging problems in dimensions higher than two

on the double random current nesting field we relate the planar random current representation introduced by griffiths hurst and sherman to the dimer model more precisely we provide a measure preserving map between double random currents obtained as the sum of two independent random currents on a planar graph and dimers on an associated bipartite graph we also define a nesting field for the double random current which under this map corresponds to the height function of the dimer model as applications we provide an alternative derivation of some of the bozonization rules obtained recently by dubedat and show that the spontaneous magnetization of the ising model on a planar biperiodic graph vanishes at criticality

gift wrapping based preimage computation algorithm based on a classical convex hull algorithm called gift wrapping the purpose of the paper is to provide a new algorithm for computing the vertices of a polytope called preimage roughly the set of naive digital planes containing a finite subset s of z 3 the vertices of the upper hemisphere the ones of the lower hemisphere and at last the equatorial vertices are computed independently the principle of the algorithm is based on duality and especially on the fact that the vertices of the preimage correspond to faces of the input set s or of its chords set email protected email protected 0 0 1 it allows to go from one vertex to another by gift wrapping until the whole region of interest has been explored

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

quantum field theory models on fractal spacetime i introduction and overview deux types de modeles de champs scalaires sont consideres l un a son propagateur donne par un temps propre ou une representation chaleur noyau l autre a un propagateur hierarchique la version du reseau fractal du propagateur a temps propre est une reflexion positive

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

new monte carlo method for the self avoiding walk we introduce a new monte carlo algorithm for the self avoiding walk saw and show that it is particularly efficient in the critical region long chains we also introduce new and more efficient statistical techniques we employ these methods to extract numerical estimates for the critical parameters of the saw on the square lattice we find 2 63820 0 00004 0 00030 1 352 0 006 0 025 v 0 7590 0 0062 0 0042 where the first error bar represents systematic error due to corrections to scaling subjective 95 confidence limits and the second bar represents statistical error classical 95 confidence limits these results are based on saws of average length 166 using 340 hours cpu time on a cdc cyber 170 730 we compare our results to previous work and indicate some directions for future research

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

lebowitz inequalities for ashkin teller systems abstract we consider the ashkin teller model with negative four spin coupling but still in the region where the ground state is ferromagnetic we establish the standard lebowitz inequality as well as the extension that is necessary to prove a divergent susceptibility

the classical limit of ultralocal scalar fields the classical limit of the quantum theory of ultralocal scalar fields is approached with the help of coherent state techniques in the limit 0 it is demonstrated that the coherent state expectation value of the heisenberg field operators coincide with the solutions of the corresponding classical canonical equations of motion

numerical solutions of the classical blasius flat plate problem abstract this paper presents a numerical study of the non linear differential equation af ff 0 where a prime denotes differentiation with respect to the similarity variable and a is a parameter for a 1 and a 2 this equation is a form of the blasius relation for the flat plate flow in fluid mechanics several numerical solution are obtained using a runge kutta algorithm for high order initial value problems for 1 a 2

laguerre collocation method for the flow and heat transfer due to a permeable stretching surface embedded in a porous medium with a second order slip and viscous dissipation a numerical method is given for studying the effect of viscous dissipation on the steady flow with heat transfer of newtonian fluid towards a permeable stretching surface embedded in a porous medium with a second order slip the governing nonlinear partial differential equations are converted into nonlinear ordinary differential equations by using similarity variables the proposed method is based on replacement of the unknown function by truncated series of well known laguerre expansion of functions an approximate formula of the derivative is introduced special attention is given to study the convergence analysis and derive an upper bound of the error of the presented approximate formula the introduced method converts the proposed equation by means of collocation points to a system of algebraic equations with laguerre coefficients thus by solving this system of equations the laguerre coefficients are obtained graphically results are shown for non dimensional velocities and temperature the effects of the porous parameter the suction injection parameter eckert number first and second order velocity slip parameter and the prandtl number on the flow and temperature profiles are presented moreover the local skin friction and nusselt numbers are presented a comparison of numerical results is made with the earlier published results under limiting cases

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

on a certain boundary value problem arising in shrinking sheet flows abstract this paper presents an analysis of the boundary value problem resulting from the magnetohydrodynamic mhd viscous flow influenced by a shrinking sheet with suction for the cases of two dimensional m 1 and axisymmetric m 2 shrinking the influences of the parameter m as well as the effects of suction parameter s and hartmann number m 2 on similar entrainment velocity f and flow characteristics are studied to this purpose the resulting nonlinear ordinary differential equation is solved numerically using the 4th order runge kutta method in combination with a shooting procedure the obtained results elucidate reliability and efficiency of the technique from which interesting features between the skin friction coefficient f 0 and the entrainment velocity f as function of the mass transfer parameter s can also be obtained

on the motion of the first lee yang zero in the d dimensional ferromagnetic ising and 4 systems the motion of the first lee yang zero when the lattice size tends to infinity is studied in particular a power law behavior at the critical point which is distinct from those in noncritical points is elucidated

on limit theorems for the bivariate magnetization energy variable at the critical point the limiting magnetization energy bivariate variable is studied for ising ferromagnets at the critical point the factorization property of the limiting bivariate moment generating function is shown to be intimately connected to critical point exponent inequalities and to the behaviour of the scaling limit near and at the critical point the validity of this can be deduced from the study of the second and the fourth magnetization cumulants at zero external field the limiting bivariate variable is exactly calculated at the critical point for the curie weiss model mf and for the edge of a two dimensional ising ferromagnet wrapped on a cylinder it is shown that the mean field case leads to a non gaussian limiting distribution in contradistinction with the particular ising model we consider for which we obtain a product of two gaussian probability distributions

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

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

interface surface tension and reentrant pinning transition in the 2d ising model we develop a new way to look at the high temperature representation of the ising model up to the critical temperature and obtain a number of interesting consequences in the two dimensional case it is possible to use these tools to prove results on phase separation lines in the whole phase coexistence regime by way of a duality transformation we illustrate the power of these techniques by studying an ising model with a boundary magnetic field in which a reentrant pinning transition takes place more precisely we show that the typical configurations of the model can be described at the macroscopic level by interfaces which are solutions of the corresponding thermodynamic variational problem this variational problem is solved explicitly there exist values of the boundary magnetic field and temperatures 0 t 2 but is pinned for t 1 t 3 it is not pinned an important property of the surface tension which is used in this paper is the sharp triangle inequality about which we report some new results the techniques used in this work are robust and can be used in a variety of different situations

lace expansion for the ising model the lace expansion has been a powerful tool for investigating mean field behavior for various stochastic geometrical models such as self avoiding walk and percolation above their respective upper critical dimension in this paper we prove the lace expansion for the ising model that is valid for any spin spin coupling for the ferromagnetic case we also prove that the expansion coefficients obey certain diagrammatic bounds that are similar to the diagrammatic bounds on the lace expansion coefficients for self avoiding walk as a result we obtain gaussian asymptotics of the critical two point function for the nearest neighbor model with d gg 4 and for the spread out model with d 4 and l gg 1 without assuming reflection positivity

the triangle condition for percolation aizenman and newman introduced an unverified condition the triangle condition which has been shown to imply that a number of percolation critical exponents take their mean field values and which is expected to hold above six dimensions for nearest neighbour percolation we prove that the triangle condition is satisfied in sufficiently high dimensions for the nearest neighbour model and above six dimensions for a class of spread out models the proof uses an expansion which is related to the lace expansion for self avoiding walk percolation is a simple probabilistic model for which many interesting problems remain unsolved the basic objects in percolation are random graphs on an infinite lattice typically there is a critical density for the graph edges known as bonds below which there is zero probability that a fixed point in the lattice is part of an infinite connected graph but above which this probability is strictly positive this abrupt change in behaviour has been used in chemistry and statistical physics to model phase transitions in a variety of physical systems such as fluid flow through a porous medium hence the name percolation random resistor networks and the gelation of branched polymers for an introduction see 15 20 to define the models we are interested in we consider the of dimensional integer lattice z whose elements are referred to as sites and the set of pairs b x y of distinct sites the bonds to each bond b is associated an independent bernoulli random variable nb9 with nb 1 with probability p jb and nb 0 with probability p jh the jb are fixed and z invariant and p is a parameter the nearest neighbour model is defined by taking jb 1 for those b x y such that x y i 1 and jb 0 otherwise expectation with respect to the joint distribution of the nb is denoted by if nb 1 we say b is occupied and otherwise b is vacant given a bond configuration nb two sites x and y are said to be connected if there is a path from x to y which consists of occupied bonds the set of sites which are connected to x is called the connected cluster of x and is denoted c x the probability that x is connected to y is written tp x y 1 ip x y i x and y are connected i y e c x p where i denotes the indicator function denoting by c x the number of sites in c x the expected cluster size or susceptibility is given by

viscous flow and heat transfer over a nonlinearly stretching sheet this paper presents a numerical analysis for flow and heat transfer in a viscous fluid over a nonlinear stretching sheet by employing a novel numerical procedure and two cases are studied namely i the sheet with constant surface temperature cst case and ii the sheet with prescribed surface temperature pst case the governing partial differential equations are converted into highly nonlinear ordinary differential equations by a similarity transformation the variations of dimensionless surface temperature dimensionless surface temperature gradient as well as flow and heat transfer characteristics with the governing parameters are graphed and tabulated

signs of the ising model ursell functions it is proven that the ursell functionsu2k of the ising model have the conjectured signs 1 k 1u2k 0 the proof is based on aizenman s random current representation and combinatorics

critical dynamics and fluctuations for a mean field model of cooperative behavior the main objective of this paper is to examine in some detail the dynamics and fluctuations in the critical situation for a simple model exhibiting bistable macroscopic behavior the model under consideration is a dynamic model of a collection of anharmonic oscillators in a two well potential together with an attractive mean field interaction the system is studied in the limit as the number of oscillators goes to infinity the limit is described by a nonlinear partial differential equation and the existence of a phase transition for this limiting system is established the main result deals with the fluctuations at the critical point in the limit as the number of oscillators goes to infinity it is established that these fluctuations are non gaussian and occur at a time scale slower than the noncritical fluctuations the method used is based on the perturbation theory for markov processes developed by papanicolaou stroock and varadhan adapted to the context of probability measure valued processes

self avoiding walk in five or more dimensions i the critical behaviour we use the lace expansion to study the standard self avoiding walk in thed dimensional hypercubic lattice ford 5 we prove that the numbercn ofn step self avoiding walks satisfiescn a n where is the connective constant i e 1 and that the mean square displacement is asymptotically linear in the number of steps i e v 1 2 a bound is obtained forcn x the number ofn step self avoiding walks ending atx the correlation length is shown to diverge asymptotically like z 1 2 the critical two point function is shown to decay at least as fast as x 2 and its fourier transform is shown to be asymptotic to a multiple ofk 2 ask 0 i e 0 we also prove that the scaling limit is gaussian in the sense of convergence in distribution to brownian motion the infinite self avoiding walk is constructed in this paper we prove these results assuming convergence of the lace expansion the convergence of the lace expansion is proved in a companion paper

emergent planarity in two dimensional ising models with finite range interactions the known pfaffian structure of the boundary spin correlations and more generally order disorder correlation functions is given a new explanation through simple topological considerations within the model s random current representation this perspective is then employed in the proof that the pfaffian structure of boundary correlations emerges asymptotically at criticality in ising models on mathbb z 2 with finite range interactions the analysis is enabled by new results on the stochastic geometry of the corresponding random currents the proven statement establishes an aspect of universality seen here in the emergence of fermionic structures in two dimensions beyond the solvable cases