ornstein zernike theory for finite range ising models above t c we derive a precise ornstein zernike asymptotic formula for the decay of the two point function 0 x in the general context of finite range ising type models on d the proof relies in an essential way on the a priori knowledge of the strict exponential decay of the two point function and by the sharp characterization of phase transition due to aizenman barsky and fernandez goes through in the whole of the high temperature region c as a byproduct we obtain that for every c the inverse correlation length is an analytic and strictly convex function of direction

fluctuation theory of connectivities for subcritical random cluster models we develop a fluctuation theory of connectivities for subcritical random cluster models the theory is based on a comprehensive nonperturbative probabilistic description of long connected clusters in terms of essentially one dimensional chains of irreducible objects statistics of local observables for example displacement over such chains obey classical limit laws and our construction leads to an effective random walk representation of percolation clusters the results include a derivation of a sharp ornstein zernike type asymptotic formula for two point functions a proof of analyticity and strict convexity of inverse correlation length and a proof of an invariance principle for connected clusters under diffusive scaling in two dimensions duality considerations enable a reformulation of these results for supercritical nearest neighbor random cluster measures in particular for nearest neighbor potts models in the phase transition regime accordingly we prove that in two dimensions potts equilibrium crystal shapes are always analytic and strictly convex and that the interfaces between different phases are always diffusive thus no roughening transition is possible in the whole regime where our results apply our results hold under an assumption of exponential decay of finite volume wired connectivities assumption 1 2 below in rectangular domains that is conjectured to hold in the whole subcritical regime the latter is known to be true in any dimensions when q 1 q 2 and when q is sufficiently large in two dimensions assumption 1 2 holds whenever there is an exponential decay of connectivities in the infinite volume measure by duality this includes all supercritical nearest neighbor potts models with positive surface tension between ordered phases

on the gibbs states of the noncritical potts model on z 2 we prove that all gibbs states of the q state nearest neighbor potts model on mathbb z 2 below the critical temperature are convex combinations of the q pure phases in particular they are all translation invariant to achieve this goal we consider such models in large finite boxes with arbitrary boundary condition and prove that the center of the box lies deeply inside a pure phase with high probability our estimate of the finite volume error term is of essentially optimal order which stems from the brownian scaling of fluctuating interfaces the results hold at any supercritical value of the inverse temperature beta beta c q log left 1 sqrt q right

analyticity properties and a convergent expansion for the glueball mass and dispersion curve of strongly coupled euclidean 2 1 lattice gauge theories for certain strongly coupled 2 g2 0 and small euclidean z3 lattice gauge theories we show that the glueball mass m associated with the truncated plaquette plaquette correlation function admits the representation m 4 ln r r n 0bn n is a gauge group representation dependent function analytic at 0 a finite algorithm is given for determining bn bn depends on a finite number of the 0 taylor series coefficients of the finite lattice correlation function at a finite number of points increasing with n of z3

asymptotics of even even correlations in the ising model we consider finite range ferromagnetic ising models on mathbb z d in the regime beta beta mathrm scriptscriptstyle c we analyze the behavior of the prefactor to the exponential decay of mathrm cov sigma a sigma b for arbitrary finite sets a and b of even cardinality as the distance between a and b diverges

large deviations for the 2d ising model a lower bound without cluster expansions we show that a lower large deviation bound for the block spin magnetization in the 2d ising model can be pushed all the way forward toward its correct wulff value for all c

comment on star star relations in statistical mechanics and elliptic gamma function identities we prove a recently conjectured star star relation which plays the role of an integrability condition for a class of 2d ising type models with multicomponent continuous spin variables namely we reduce this relation to an identity for elliptic gamma functions previously obtained by rains

one particle subspaces in the stochastic xy model we study the spectrum of the generatorh of the glauber dynamics for a model of planar rotators on a lattice in the case of a high temperature 1 we construct two so called one particle subspacesh forh and describe the spectrum of the generator in these subspaces as a consequence we find time asymptotics of the correlations for the glauber dynamics

large deviations and continuum limit in the 2d ising model we study the 2d ising model in a rectangular box l of linear size o l we determine the exact asymptotic behaviour of the large deviations of the magnetization p t2 l ae t when l 1 for values of the parameters of the model corresponding to the phase coexistence region where the order parameter m is strictly positive we study in particular boundary eects due to an arbitrary real valued boundary magnetic field using the self duality of the model a large part of the analysis consists in deriving properties of the covariance function hae 0 ae t i as t 1 at dual values of the parameters of the model to do this analysis we establish new results about the high temperature representation of the model these results are valid for dimensions d 2 and up to the critical temperature they give a complete non perturbative exposition of the high temperature representation we then study the gibbs measure conditioned by p t2 l ae t im l l l ic with 0 c 1 4 and im m m we construct the continuum limit of the model and describe the limit by the solutions of a variational problem of isoperi metric type

crossing random walks and stretched polymers at weak disorder we consider a model of a polymer in d 1 constrained to join 0 and a hyperplane at distance n the polymer is subject to a quenched nonnegative random environment alternatively the model describes crossing random walks in a random potential see zerner ann appl probab 8 1998 246 280 or chapter 5 of sznitman brownian motion obstacles and random media 1998 springer for the original brownian motion formulation it was recently shown ann probab 36 2008 1528 1583 probab theory related fields 143 2009 615 642 that in such a setting the quenched and annealed free energies coincide in the limit n when d 3 and the temperature is sufficiently high we first strengthen this result by proving that under somewhat weaker assumptions on the distribution of disorder which in particular enable a small probability of traps the ratio of quenched and annealed partition functions actually converges we then conclude that in this case the polymer obeys a diffusive scaling with the same diffusivity constant as the annealed model

convexity properties of the surface tension and equilibrium crystals we study the thermodynamic limit of the orientation dependent surface tension under general conditions which we show to hold true for a large class of lattice systems we prove that the limit exists and that it satisfies some convexity properties related to the pyramidal inequality introduced by r l dobrushin and s b shlosman 1 we discuss some consequences of these results for the equilibrium crystal shape

a local limit theorem for triple connections in subcritical bernoulli percolation we prove a local limit theorem for the probability of a site to be connected by disjoint paths to three points in subcritical bernoulli percolation on mathbb z d d geq2 in the limit where their distances tend to infinity

subcritical percolation with a line of defects we consider the bernoulli bond percolation process pp p on the nearest neighbor edges of zd which are open independently with probability p pc except for those lying on the first coordinate axis for which this probability is p define p p limn n 1logpp p 0 ne1 and p p p we show that there exists p c p c p d such that p p p if p p c moreover p c p 2 p c p 3 p and p c p d p for d 4 we also analyze the behavior of p p p as p p c in dimensions d 2 3 finally we prove that when p p c the following purely exponential asymptotics holds pp p 0 ne1 de p p n 1 o 1 for some constant d d p p uniformly for large values of n this work gives the first results on the rigorous analysis of pinning type problems that go beyond the effective models and don t rely on exact computations

quasi monte carlo methods for two stage stochastic mixed integer programs we consider randomized qmc methods for approximating the expected recourse in two stage stochastic optimization problems containing mixed integer decisions in the second stage it is known that the second stage optimal value function is piecewise linear quadratic with possible kinks and discontinuities at the boundaries of certain convex polyhedral sets this structure is exploited to provide conditions implying that first and higher order terms of the integrand s anova decomposition math comp 79 2010 953 966 have mixed weak first order partial derivatives this leads to a good smooth approximation of the integrand and hence to good convergence rates of randomized qmc methods if the effective superposition dimension is low

four particle decay of the bethe salpeter kernel in the high temperature ising model in this article we study the four particle decay of the bethe salpeter b s kernel for the high temperature ising model we use the hyperplane decoupling method t spencer commun math phys 44 143 1975 r s schor nucl phys b 222 71 1983 to prove exponential decay in a set of variables particularly adapted to the methods of spencer and zirilli commun math phys 49 1 1976 for the analysis of scattering and bound states in qft transcribed to lattice theories by auil and barata ann henri poincare 2 1065 2001 we study arbitrary derivatives of the general n point correlation functions with respect to the interpolating variables and we are able to obtain in some cases information about the third derivatives of the b s kernel as a later consequence we have two body asymptotic completeness for the massive euclidean lattice field theory implemented by this model this allows us to analyze the ornstein zernike behavior of four point functions related to the specific heat of th

approximation of subadditive functions and convergence rates in limiting shape results for a nonnegative subadditive function h on mathbb z d with limiting approximation g x lim n h nx n it is of interest to obtain bounds on the discrepancy between g x and h x typically of order x nu with nu 1 for certain subadditive h x particularly those which are expectations associated with optimal random paths from 0 to x in a somewhat standardized way a more natural and seemingly weaker property can be established every x is in a bounded multiple of the convex hull of the set of sites satisfying a similar bound we show that this convex hull property implies the desired bound for all x applications include rates of convergence in limiting shape results for first passage percolation standard and oriented and longest common subsequences and bounds on the error in the exponential decay approximation to the off axis connectivity function for subcritical bernoulli bond percolation on the integer lattice

spin chains and gustafson s integrals gustafson s integrals are multidimensional generalizations of the classical mellin barnes integrals we show that some of these integrals arise from relations between matrix elements in sklyanin s representation of separated variables in spin chain models we also present several new integrals

analyticity properties and a convergent expansion for the inverse correlation length of the high temperatured dimensional ising model we show that the inverse correlation lengthm mass of the fundamental particle of the associated lattice quantum field theory of the spin spin correlation function s x s y x y ez d of thed dimensional ising model admits the representation m beta ln beta r beta for small inverse temperatures 0 r is ad dependent function analytic at 0 c n the nth 0 taylor series coefficient of r can be computed explicitly from the zd limit of a finite number of finite lattice a spin spin correlation functions s0sx t afor a finite number ofx x1 x2 xd x i d 1 xi 0 such that for each e 0 m is analytic similar results are also obtained for the dispersion curve p p 0 m pe d 1 of the fundamental particle of the associated lattice quantum field theory

entropic repulsion of an interface in an external field we consider an interface above an attractive hard wall in the complete wetting regime and submitted to the action of an external increasing convex potential and study its delocalization as the intensity of this potential vanishes our main motivation is the analysis of critical prewetting which corresponds to the choice of a linear external potential we also present partial results on critical prewetting in the two dimensional ising model as well as a few weak results on pathwise estimates for the pure wetting problem for effective interface models

efficient computation of option prices and greeks by quasi monte carlo method with smoothing and dimension reduction discontinuities and high dimensionality are common in the problems of pricing and hedging of derivative securities both factors have a tremendous impact on the accuracy of the quasi monte carlo qmc method an ideal approach to improve the qmc method is to transform the functions to make them smoother and having smaller effective dimension this paper develops a two step procedure to tackle the challenging problems of both the discontinuity and the high dimensionality concurrently in the first step we adopt the smoothing method to remove the discontinuities of the payoff function improving the smoothness in the second step we propose a general dimension reduction method called the cqr method to reduce the effective dimension such that the better quality of qmc points in their initial dimensions can be fully exploited the cqr method is based on the combination of the k means clustering algorithm and the qr decomposition the k means clustering algorithm a classical algorithm of machine lear

a new solution of the star triangle relation we obtain a new solution to the star triangle relation for an ising type model with two kinds of spin variables at each lattice site taking continuous real values and arbitrary integer values respectively the boltzmann weights are manifestly real and positive they are expressed through the euler gamma function and depend on sums and differences of spins at the ends of an edge of the lattice

classification of integrable equations on quad graphs the consistency approach a classification of discrete integrable systems on quad graphs i e on surface cell decompositions with quadrilateral faces is given the notion of integrability laid in the basis of the classification is the three dimensional consistency this property yields among other features the existence of the discrete zero curvature representation with a spectral parameter for all integrable systems of the obtained exhaustive list the so called three leg forms are found this establishes lagrangian and symplectic structures for these systems and the connection to discrete systems of the toda type on arbitrary graphs generalizations of these ideas to the three dimensional integrable systems and to the quantum context are also discussed

rarefied elliptic hypergeometric functions abstract two exact evaluation formulae for multiple rarefied elliptic beta integrals related to the simplest lens space are proved they generalize evaluations of the type i and ii elliptic beta integrals attached to the root system c n in a special n 1 case the simplest p 0 limit is shown to lead to a new class of q hypergeometric identities symmetries of a rarefied elliptic analogue of the euler gauss hypergeometric function are described and the respective generalization of the hypergeometric equation is constructed some extensions of the latter function to c n and a n root systems and corresponding symmetry transformations are considered an application of the rarefied type ii c n elliptic hypergeometric function to some eigenvalue problems is briefly discussed

the smoothing effect of integration in r d and the anova decomposition this paper studies the anova decomposition of a d variate function f dened on the whole of r d where f is the maximum of a smooth function and zero or f could be the absolute value of a smooth function our study is motivated by option pricing problems we show that under suitable conditions all terms of the anova decomposition except the one of highest order can have unlimited smoothness in particular this is the case for arithmetic asian options with both the standard and brownian bridge constructions of the brownian motion 2010 mathematics subject classication primary 41a63 41a99 secondary 65d30

ornstein zernike theory for the bernoulli bond percolation on mathbb z d we derive a precise ornstein zernike asymptotic formula for the decay of the two point function p p 0 x of the bernoulli bond percolation on the integer lattice z d in any dimension d 2 in any direction x and for any subcritical value of p p c d

hyperbolic beta integrals abstract hyperbolic beta integrals are analogues of euler s beta integral in which the role of euler s gamma function is taken over by ruijsenaars hyperbolic gamma function they may be viewed as q q bibasic analogues of the beta integral in which the two bases q and q are interrelated by modular inversion and they entail q analogues of the beta integral for q 1 the integrals under consideration are the hyperbolic analogues of the ramanujan integral the askey wilson integral and the nassrallah rahman integral we show that the hyperbolic nassrallah rahman integral is a formal limit case of spiridonov s elliptic nassrallah rahman integral

a simple inductive approach to the problem of convergence of cluster expansions of polymer models we explain a simple inductive method for the analysis of the convergence of cluster expansions taylor expansions mayer expansions for the partition functions of polymer models we give a very simple proof of the dobrushin koteck preiss criterion and formulate a generalization usable for situations where a successive expansion of the partition function has to be used

rigorous probabilistic analysis of equilibrium crystal shapes the rigorous microscopic theory of equilibrium crystal shapes has made enormous progress during the last decade we review here the main results that have been obtained both in two and higher dimensions in particular we describe how the phenomenological wulff and winterbottom constructions can be derived from the microscopic description provided by the equilibrium statistical mechanics of lattice gases we focus on the main conceptual issues and describe the central ideas of the existing approaches

transformations of elliptic hypergeometric integrals we prove a pair of transformations relating elliptic hypergeometric integrals of different dimensions corresponding to the root systems bc n and a n as a special case we recover some integral identities conjectured by van diejen and spiridonov for bc n we also consider their type ii integral their proof of that integral together with our transformation gives rise to pairs of adjoint integral operators a different proof gives rise to pairs of adjoint difference operators these allow us to construct a family of biorthogonal abelian functions generalizing the koornwinder polynomials and satisfying the analogues of the macdonald conjectures finally we discuss some transformations of type ii style integrals in particular we find that adding two parameters to the type ii integral gives an integral invariant under an appropriate action of the weyl group e 7

ornstein zernike asymptotics for a general two particle lattice operator we study the asymptotic behavior of correlations for a general two particle operator mathcal t acting on the hilbert space ell 2 mathbb z d times mathbb z d for all dimension d 1 2 mathcal t is written as the sum of a main term and a small interacting term a form which appears in many problems if the interacting term is small we give a complete description of the asymptotics for large t of the correlations mathcal t t f 1 f 2 t 1 2 ldots for f 1 f 2 in some suitable class the asymptotics is of the ornstein zernike type i e exponential with a power law factor which is t d for d 3 but for d 1 2 it can be anomalous and is determined by the interacting term

first order analytic difference equations and integrable quantum systems we present a new solution method for a class of first order analytic difference equations the method yields explicit minimal solutions that are essentially unique special difference equations give rise to minimal solutions that may be viewed as generalized gamma functions of hyperbolic trigonometric and elliptic type euler s gamma function being of rational type we study these generalized gamma functions in considerable detail the scattering and weight functions u and w functions associated to various integrable quantum systems can be expressed in terms of our generalized gamma functions we obtain detailed information on these u and w functions exploiting the difference equations they satisfy

a finite volume version of aizenman higuchi theorem for the 2d ising model in the late 1970s in two celebrated papers aizenman and higuchi independently established that all infinite volume gibbs measures of the two dimensional ferromagnetic nearest neighbor ising model at inverse temperature beta geq 0 are of the form alpha mu beta 1 alpha mu beta where mu beta and mu beta are the two pure phases and 0 leq alpha leq 1 we present here a new approach to this result with a number of advantages a we obtain an optimal finite volume quantitative analogue implying the classical claim b the scheme of our proof seems more natural and provides a better picture of the underlying phenomenon c this new approach might be applicable to systems for which the classical method fails

critical region for droplet formation in the two dimensional ising model we study the formation dissolution of equilibrium droplets in finite systems at parameters corresponding to phase coexistence specifically we consider the 2d ising model in volumes of size l2 inverse temperature c and overall magnetization conditioned to take the value m l2 2m vl where c 1 is the critical temperature m m is the spontaneous magnetization and vl is a sequence of positive numbers we find that the critical scaling for droplet formation dissolution is when vl3 2l 2 tends to a definite limit specifically we identify a dimensionless parameter proportional to this limit a non trivial critical value c and a function such that the following holds for c there is a single wulff shaped droplet containing a fraction c 2 3 of the magnetization deficit and there are no other droplets beyond the scale of log l moreover and are related via a universal equation that apparently is independent of the details of the system

asymptotics of decay of correlations for lattice spin fields at high temperatures i the ising model we find the asymptotic decrease of correlations y zv 1 y in the ising model at high temperatures for the case when monomials a and b both are odd using the saddle point method we find the asymptotics of the correlations for any dimension for even monomials a b we formulate a general hypothesis about the form of the asymptotics and confirm it in two cases 1 1 and the vectory has an arbitrary direction 2 y is directed along a fixed axis and arbitrary here we use besides the saddle point method some arguments from scattering theory

spectral analysis of weakly coupled stochastic lattice ginzburg landau models we consider the relaxation to equilibrium of solutions t 0 of stochastic dynamical langevin equations with white noise and weakly coupled ginzburg landau interactions using a feynman kac formula which relates stochastic expectations to correlation functions of a spatially non local imaginary time quantum field theory we obtain results on the joint spectrum of h where h is the self adjoint positive generator of the semi group associated with the dynamics and p j j 1 d are the self adjoint generators of the group of lattice spatial translations we show that the low lying energy momentum spectrum consists of an isolated one particle dispersion curve and for the mass spectrum energy momentum at zero momentum besides this isolated one particle mass we show using a bethe salpeter equation the existence of an isolated two particle bound state if the coefficient of the quartic term in the polynomial of the ginzburg landau interaction is negative and d 1 2 otherwise there is no two particle bound state asymptotic values for the masses are obtained

phase separation in random cluster models iii circuit regularity we study the droplet that results from conditioning the subcritical fortuin kasteleyn random cluster model on the presence of an open circuit encircling the origin and enclosing an area of at least or exactly n 2 in this paper we prove that the resulting circuit is highly regular we define a notion of a regeneration site in such a way that for any such site v the circuit cuts through the radial line segment through v only at v we show that provided that the conditioned circuit is centred at the origin in a natural sense the set of regeneration sites reaches into all parts of the circuit with maximal distance from one such site to the next being at most logarithmic in n with high probability the result provides a flexible control on the conditioned circuit that permits the use of surgical techniques to bound its fluctuations and as such it plays a crucial role in the derivation of bounds on the local fluctuation of the circuit carried out in hammond arxiv 1001 1527 2009 arxiv 1001 1528 2009

new solutions of the star triangle relation with discrete and continuous spin variables a new solution to the star triangle relation is given for an ising type model of interacting spins containing integer and real valued components boltzmann weights of the model are given in terms of the lens elliptic gamma function and are based on yamazaki s recently obtained solution of the star star relation the star triangle relation given here implies seiberg duality for the 4 d index of the su 2 quiver gauge theory and the corresponding two component spin case of the star star relation of yamazaki a proof of the star triangle relation is given resulting in a new elliptic hypergeometric summation integration identity the star triangle relation in this paper contains the master solution of bazhanov and sergeev as a special case two other limiting cases are considered one of which gives a new star triangle relation in terms of ratios of infinite q products while the other case gives a new way of deriving a star triangle relation that was previously obtained by the author

gaussian fluctuations of connectivities in the subcritical regime of percolation we consider thed dimensional bernoulli bond percolation model and prove the following results for allp pc the finite volume conditional measures defined by requiring the existence of a dual path between opposing faces of the boundary converge in the infinite volume limit to the standard bernoulli measure

variational principles for circle patterns and koebe s theorem we prove existence and uniqueness results for patterns of circles with prescribed intersection angles on constant curvature surfaces our method is based on two new functionals one for the euclidean and one for the hyperbolic case we show how colin de verdiere s bragger s and rivin s functionals can be derived from ours

spectral properties of weakly coupled landau ginzburg stochastic models we study the existence of bound states in the generator of the stochastic dynamics associated to weakly coupled lattice landau ginzburg models by analyzing the bethe salpeter kernel in the ladder approximation these states are shown to exist if the polynomial interaction has a negative quartic term and the lattice dimension is smaller than 3 asymptotic values for the masses are also obtained giving precise relaxation rates for even correlations

phase separation in random cluster models i uniform upper bounds on local deviation this is the first in a series of three papers that addresses the behaviour of the droplet that results in the percolating phase from conditioning the planar fortuin kasteleyn random cluster model on the presence of an open dual circuit 0 encircling the origin and enclosing an area of at least or exactly n2 by the fortuin kasteleyn representation the model is a close relative of the droplet formed by conditioning the potts model on an excess of spins of a given type we consider local deviation of the droplet boundary measured in a radial sense by the maximum local roughness mlr 0 this being the maximum distance from a point in the circuit 0 to the boundary of the circuit s convex hull and in a longitudinal sense by what we term maximum facet length mlf 0 namely the length of the longest line segment of which the polygon is formed the principal conclusion of the series of papers is the following uniform control on local deviation that there are constants 0 c c such that the conditional probability that the normalized quantity n 1 3 log n 2 3mlr lies in the interval c c tends to 1 in the high n limit and that the same statement holds for n 2 3 log n 1 3 mlf in this way we confirm the anticipated n1 3 scaling of maximum local roughness and provide a sharp logarithmic power law correction this local deviation behaviour occurs by means of locally gaussian effects constrained globally by curvature and we believe that it arises in many radially defined stochastic interface models including growth models belonging to the kardar parisi zhang universality class

random walks in quenched i i d space time random environment are always a s diffusive we consider a general model of discrete time random walk x t on the lattice 1 in a random environment t x t x 1 with i i d components t x previous results on the a s validity of the central limit theorem for the quenched model required a small stochasticity condition in this paper we show that the result holds provided only that an obvious non degeneracy condition is met the proof is based on the analysis of a suitable generating function which allows to estimate l 2 norms by contour integrals

mixing properties and exponential decay for lattice systems in finite volumes an infinite volume mixing or exponential decay property in a spin system or percolation model reflects the inability of the influence of the configuration in one region to propagate to distant regions but in some circumstances where such properties hold propagation can nonetheless occur in finite volumes endowed with boundary conditions we establish the absense sic of such propagation particularly in two dimensions in finite volumes which are simply connected under a variety of conditions mainly for the potts model and the fortuin kasteleyn fk random cluster model allowing external fields for example for the fk model in two dimensions we show that exponential decay of connectivity in infinite volume implies exponential decay in simply connected finite volumes uniformly over all such volumes and all boundary conditions and implies a strong mixing property for such volumes with certain types of boundary conditions for the potts model in two dimensions we show that exponential decay of correlations in infinite volume implies a strong mixing property in simply connected finite volumes which includes exponential decay of correlations in simply connected finite volumes uniformly over all such volumes and all boundary conditions

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

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

on decompositions of multivariate functions we present formulas that allow us to decompose a function f of d variables into a sum of 2 d terms fu indexed by subsets u of f1 dg where each term fu depends only on the variables with indices in u the decomposition depends on the choice of d commuting projections fpjg d 1 where pj f does not depend on the variable xj we present an explicit formula for fu which is new even for the anova and anchored decompositions both are special cases of the general decomposition we show that the decomposition is minimal in the following sense if f is expressible as a sum in which there is no term that depends on all of the variables indexed by the subset z then for every choice of fpjg d 1 the terms fu 0 for all subsets u containing z furthermore in a reproducing kernel hilbert space setting we give su cient conditions for the terms fu to be mutually orthogonal

high dimensional integration of kinks and jumps smoothing by preintegration abstract we show how simple kinks and jumps of otherwise smooth integrands over r d can be dealt with by a preliminary integration with respect to a single well chosen variable it is assumed that this preintegration or conditional sampling can be carried out with negligible error which is the case in particular for option pricing problems it is proven that under appropriate conditions the preintegrated function of d 1 variables belongs to appropriate mixed sobolev spaces so potentially allowing high efficiency of quasi monte carlo and sparse grid methods applied to the preintegrated problem the efficiency of applying quasi monte carlo to the preintegrated function are demonstrated on a digital asian option using the principal component analysis factorization of the covariance matrix

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

the elliptic gamma function and sl 3 z z3 abstract the elliptic gamma function is a generalization of the euler gamma function and is associated to an elliptic curve its trigonometric and rational degenerations are the jackson q gamma function and the euler gamma function respectively the elliptic gamma function appears in baxter s formula for the free energy of the eight vertex model and in the hypergeometric solutions of the elliptic qkzb equations in this paper the properties of this function are studied in particular we show that elliptic gamma functions are generalizations of automorphic forms of g sl 3 z z 3 associated to a non trivial class in h3 g z

constructing sobol sequences with better two dimensional projections direction numbers for generating sobol sequences that satisfy the so called property a in up to 1111 dimensions have previously been given in joe and kuo acm trans math software 29 2003 pp 49 57 however these sobol sequences may have poor two dimensional projections here we provide a new set of direction numbers alleviating this problem these are obtained by treating sobol sequences in d dimensions as t d sequences and then optimizing the t values of the two dimensional projections our target dimension is 21201

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

dynamic critical behavior of a swendsen wang type algorithm for the ashkin teller model we study the dynamic critical behavior of a swendsen wang type algorithm for the ashkin teller model we find that the li sokal bound on the autocorrelation time int const xc h holds along the self dual curve of the symmetric ashkin teller model and is almost but not quite sharp the ratio int c h appears to tend to infinity either as a logarithm or as a small power 0 05 p 0 12 in an appendix we discuss the problem of extracting estimates of the exponential autocorrelation time

the self dual point of the two dimensional random cluster model is critical for q 1 we prove a long standing conjecture on random cluster models namely that the critical point for such models with parameter q 1 on the square lattice is equal to the self dual point p sd q sqrt q 1 sqrt q this gives a proof that the critical temperature of the q state potts model is equal to log 1 sqrt q for all q 2 we further prove that the transition is sharp meaning that there is exponential decay of correlations in the sub critical phase the techniques of this paper are rigorous and valid for all q 1 in contrast to earlier methods valid only for certain given q the proof extends to the triangular and the hexagonal lattices as well

three body asymptotic completeness for p φ 2 models we consider weakly coupled even p 2 models that do not have a two body bound state and prove asymptotic completeness on the subspace of states with mass between 3m a and 4m b wherea andb are positive functions tending to zero with the analytic structure of the six point function integrated over the three incoming momenta shows only two landau singular manifolds plus normal thresholds associated to three particle processes

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