uniqueness of gradient gibbs measures with disorder we consider in a uniformly strictly convex potential regime two versions of random gradient models with disorder in model a the interface feels a bulk term of random fields while in model b the disorder enters through the potential acting on the gradients we assume a general distribution on the disorder with uniformly bounded finite second moments it is well known that for gradient models without disorder there are no gibbs measures in infinite volume in dimension d 2 while there are shift invariant gradient gibbs measures describing an infinite volume distribution for the gradients of the field as was shown by funaki and spohn commun math phys 185 1 36 1997 van enter and kulske proved in ann appl probab 18 1 109 119 2008 that adding a disorder term as in model a prohibits the existence of such gradient gibbs measures for general interaction potentials in d 2 in cotar and kulske ann appl probab 22 5 1650 1692 2012 we proved the existence of shift covariant random gradient gibbs measures for model a when d ge 3 the disorder is i i d and has mean zero and for model b when d ge 1 and the disorder has a stationary distribution in the present paper we prove existence and uniqueness of shift covariant random gradient gibbs measures with a given expected tilt u in mathbb r d and with the corresponding annealed measure being ergodic for model a when d ge 3 and the disordered random fields are i i d and symmetrically distributed and for model b when d ge 1 and for any stationary disorder dependence structure we also compute for both models for any gradient gibbs measure constructed as in cotar and kulske ann appl probab 22 5 1650 1692 2012 when the disorder is i i d and its distribution satisfies a poincare inequality assumption the optimal decay of covariances with respect to the averaged over the disorder gradient gibbs measure

multilevel frames for sparse tensor product spaces for au f with an elliptic differential operator a mathcal h rightarrow mathcal h and stochastic data f the m point correlation function mathcal m m u of the random solution u satisfies a deterministic equation with the m fold tensor product operator a m of a sparse tensor products of hierarchic fe spaces in mathcal h are known to allow for approximations to mathcal m m u which converge at essentially the rate as in the case m 1 i e for the deterministic problem they can be realized by wavelet type fe bases von petersdorff and schwab in appl math 51 2 145 180 2006 schwab and todor in computing 71 43 63 2003 if wavelet bases are not available we show here how to achieve the fast computation of sparse approximations of mathcal m m u for galerkin discretizations of a by multilevel frames such as bpx or other multilevel preconditioners of any standard fem approximation for a numerical examples illustrate feasibility and scope of the method

growth and zeros of meromorphic solution of some linear difference equations abstract in this paper we study growth and zeros of linear difference equations p n z f z n p 1 z f z 1 p 0 z f z f z where f z p n z p 0 z are polynomials with f p n p 0 0 and satisfy deg p n p 0 max deg p j j 0 n 1 the corresponding homogeneous equation of the above equation is also investigated

on the exponential decay of the n point correlation functions and the analyticity of the pressure the goal of this paper is to provide estimates leading to a direct proof of the exponential decay of the n point correlation functions for certain unbounded models of kac type the methods are based on estimating higher order derivatives of the solution of the witten laplacian equation on 1 forms associated with the hamiltonian of the system we also provide a formula for the taylor coefficients of the pressure that is suitable for a direct proof of the analyticity

towards a nonperturbative renormalization group analysis we prove that a certain class of convex gradient models in high dimensional spaces without the presence of a small parameter renormalizes to a free field as a consequence we establish a certain asymptotic formula for the partition function in some ways this is a realization of gawedzki and kupiainen s idea to use correlation inequalities to augment the rigorous renormalization group methods we use the more particular suggestion of spencer to use certain inequalities of brascamp and lieb and also the formulation of the correlation functions in terms of the solutions to some partial differential equations given by helffer and sjostrand

adaptive solution of operator equations using wavelet frames in adaptive wavelet methods ii beyond the elliptic case of cohen dahmen and devore found comput math 2 2002 pp 203 245 an adaptive method has been developed for solving general operator equations using a riesz basis of wavelet type for the energy space the operator equation is transformed into an equivalent matrix vector system this system is solved iteratively where the application of the infinite stiffness matrix is replaced by an adaptive approximation assuming that the stiffness matrix is sufficiently compressible i e that it can be sufficiently well approximated by sparse matrices it was proved that the adaptive method has optimal computational complexity in the sense that it converges with the same rate as the best n term approximation for the solution assuming that the latter would be explicitly available the condition concerning compressibility requires that dependent on their order the wavelets have sufficiently many vanishing moments and that they be sufficiently smooth however except on tensor product domains wavelets that satisfy this smoothness requirement are not easy to construct in this paper we write the domain or manifold on which the operator equation is posed as an overlapping union of subdomains each of them being the image under a smooth parametrization of the hypercube by lifting wavelets on the hypercube to the subdomains we obtain a em frame for the energy space with this frame the operator equation is transformed into a matrix vector system after which this system is solved iteratively by an adaptive method similar to the one from the work of cohen dahmen and devore with this approach frame elements that have sufficiently many vanishing moments and are sufficiently smooth something needed for the compressibility are easily constructed by handling additional difficulties due to the fact that a frame gives rise to an underdetermined matrix vector system we prove that this adaptive method has optimal computational complexity

on homogenization and scaling limit of some gradient perturbations of a massless free field we study the continuum scaling limit of some statistical mechanical models defined by convex hamiltonians which are gradient perturbations of a massless free field by proving a central limit theorem for these models we show that their long distance behavior is identical to a new homogenized continuum massless free field we shall also obtain some new bounds on the 2 point correlation functions of these models

relationships between characteristic functions of δnf and f abstract in this paper we obtain the accurate relationships between the characteristic functions of a finite order meromorphic function f z and its forward differences n f z where f z has two borel exceptional values

fast multipole boundary element method to calculate head related transfer functions for a wide frequency range head related transfer functions hrtfs play an important role in spatial sound localization the boundary element method bem can be applied to calculate hrtfs from non contact visual scans because of high computational complexity hrtf simulations with bem for the whole head and pinnae have only been performed for frequencies below 10khz in this study the fast multipole method fmm is coupled with bem to simulate hrtfs for a wide frequency range the basic approach of the fmm and its implementation are described a mesh with over 70 000 elements was used to calculate hrtfs for one subject with this mesh the method allowed to calculate hrtfs for frequencies up to 35khz comparison to acoustically measured hrtfs has been performed for frequencies up to 16khz showing a good congruence below 7khz simulations with an additional shoulder mesh improved the congruence in the vertical direction reduction in the mesh size by 5 resulted in a substantially worse representation of spectral cues the effec

frames and pseudo inverses abstract we point out some connections between the existing theories for frames and pseudo inverses in particular using the pseudo inverse of an unbounded operator we show how to make frame like decompositions with respect to some sequences which do not satisfy the upper frame condition as a special case this approach immediately gives nontrivial results for usual frames

finite range decomposition for families of gradient gaussian measures abstract let a family of gradient gaussian vector fields on z d be given we show the existence of a uniform finite range decomposition of the corresponding covariance operators that is the covariance operator can be written as a sum of covariance operators whose kernels are supported within cubes of diameters l k in addition we prove natural regularity for the subcovariance operators and we obtain regularity bounds as we vary within the given family of gradient gaussian measures

wavelet galerkin schemes for boundary integral equations implementation and quadrature in the present paper we consider the fully discrete wavelet galerkin scheme for the fast solution of boundary integral equations in three dimensions it produces approximate solutions within discretization error accuracy offered by the underlying galerkin method at a computational expense that stays proportional to the number of unknowns we focus on algorithmical details of the scheme in particular on numerical integration of relevant matrix coefficients we illustrate the proposed algorithm by numerical results

reproducing pairs and the continuous nonstationary gabor transform on lca groups in this paper we introduce and investigate the concept of reproducing pairs as a generalization of continuous frames reproducing pairs yield a bounded analysis and synthesis process while the frame condition can be omitted at both stages moreover we will investigate certain continuous frames resp reproducing pairs on lca groups which can be described as a continuous version of nonstationary gabor systems and state sufficient conditions for these systems to form a continuous frame resp reproducing pair as a byproduct we identify the structure of the frame operator resp resolution operator we will apply our results to systems generated by a unitary action of a subset of the affine weyl heisenberg group in this setup will also serve as a nontrivial example of a system for which whereas continuous frames exist no dual system with the same structure exists even if we drop the frame property

pde with random coefficients and euclidean field theory in this paper a new proof of an identity of giacomin olla and spohn is given the identity relates the 2 point correlation function of a euclidean field theory to the expectation of the green s function for a pde with random coefficients the euclidean field theory is assumed to have convex potential an inequality of brascamp and lieb therefore implies gaussian bounds on the fourier transform of the 2 point correlation function by an application of results from random pde the previously mentioned identity implies pointwise gaussian bounds on the 2 point correlation function

correlation at low temperature ii asymptotics the present paper is a continuation of ref 4 where the truncated two point correlation function for a class of lattice spin systems was proved to have exponential decay at low temperature under a weak coupling assumption in this paper we compute the asymptotics of the correlation function as the temperature goes to zero this paper thus extends ref 3 in two directions the hamiltonian function is allowed to have several local minima other than a unique global minimum and we do not require translation invariance of the hamiltonian function we are in particular able to handle spin systems on a general lattice

on the correlation for kac like models in the convex case the aim of this paper is to stu the behavior asm tends to of a family of measures exp m x dx m on m where m is a potential on m which is a perturbation in a suitable sense of the harmonic potential j x j 2

a renormalisation group method i gaussian integration and normed algebras this paper is the first in a series devoted to the development of a rigorous renormalisation group method for lattice field theories involving boson fields fermion fields or both our immediate motivation is a specific model involving both boson and fermion fields which arises as a representation of the continuous time weakly self avoiding walk in this paper we define normed algebras suitable for a renormalisation group analysis and develop methods for performing analysis on these algebras we also develop the theory of gaussian integration on these normed algebras and prove estimates for gaussian integrals the concepts and results developed here provide a foundation for the continuation of the method presented in subsequent papers in the series

existence of finite order meromorphic solutions as a detector of integrability in difference equations abstract the existence of sufficiently many finite order in the sense of nevanlinna meromorphic solutions of a difference equation appears to be a good indicator of integrability it is shown that out of a large class of second order difference equations the only equation that can admit a sufficiently general finite order meromorphic solution is the difference painleve ii equation the proof given relies on estimates obtained by arguments related to singularity confinement the existence of meromorphic solutions of a general class of first order difference equations is also proven by a simple method based on banach s fixed point theorem

canonical forms of unconditionally convergent multipliers multipliers are operators that combine frame like analysis a multiplication with a fixed sequence called the symbol and synthesis they are very interesting mathematical objects that also have a lot of applications for example in acoustical signal processing it is known that bounded symbols and bessel sequences guarantee unconditional convergence in this paper we investigate necessary and equivalent conditions for the unconditional convergence of multipliers in particular we show that under mild conditions unconditionally convergent multipliers can be transformed by shifting weights between symbol and sequence into multipliers with symbol 1 and bessel sequences called multipliers in canonical form

strict convexity of the free energy for a class of non convex gradient models we consider a gradient interface model on the lattice with interaction potential which is a non convex perturbation of a convex potential we show using a one step multiple scale analysis the strict convexity of the surface tension at high temperature this is an extension of funaki and spohn s result 8 where the strict convexity of potential was crucial in their proof

finite range decompositions of positive definite functions abstract we give sufficient conditions for a positive definite function to admit decomposition into a sum of positive definite functions which are compactly supported within disks of increasing diameters l n more generally we consider positive definite bilinear forms f v f f defined on c 0 we say v has a finite range decomposition if v can be written as a sum v g n of positive definite bilinear forms g n such that g n f g 0 when the supports of the test functions f g are separated by a distance greater or equal to l n we prove that such decompositions exist when v is dual to a bilinear form b 2 where b is a vector valued partial differential operator satisfying some regularity conditions

the short distance behavior of φ 4 3 we consider the 3 4 quantum field theory on a torus and study the short distance behavior we reproduce the standard result that the singularities can be removed by a simple mass renormalization for the resulting model we give anl p bound on the short distance regularity of the correlation functions to obtain these results we develop a systematic treatment of the generating functional for correlations using a renormalization group method incorporating background fields

motion by mean curvature from the ginzburg landau interface model we consider the scalar field t with a reversible stochastic dynamics which is defined by the standard dirichlet form relative to the gibbs measure with formal energy the potential v is even and strictly convex we prove that under a suitable large scale limit the t field becomes deterministic such that locally its normal velocity is proportional to its mean curvature except for some anisotropy effects as an essential input we prove that for every tilt there is a unique shift invariant ergodic gibbs measure for the field

mayer expansions and the hamilton jacobi equation we review the derivation of wilson s differential equation in infinitely many variables which describes the infinitesimal change in an effective potential of a statistical mechanical model or quantum field theory when an infinitesimal integration out is performed we show that this equation can be solved for short times by a very elementary method when the initial data are bounded and analytic the resulting series solutions are generalizations of the mayer expansion in statistical mechanics the differential equation approach gives a remarkable identity for connected parts and precise estimates which include criteria for convergence of iterated mayer expansions applications include the yukawa gas in two dimensions past the 4 threshold and another derivation of some earlier results of gopfert and mack

a renormalisation group method v a single renormalisation group step this paper is the fifth in a series devoted to the development of a rigorous renormalisation group method applicable to lattice field theories containing boson and or fermion fields and comprises the core of the method in the renormalisation group method increasingly large scales are studied in a progressive manner with an interaction parametrised by a field polynomial which evolves with the scale under the renormalisation group map in our context the progressive analysis is performed via a finite range covariance decomposition perturbative calculations are used to track the flow of the coupling constants of the evolving polynomial but on their own perturbative calculations are insufficient to control error terms and to obtain mathematically rigorous results in this paper we define an additional non perturbative coordinate which together with the flow of coupling constants defines the complete evolution of the renormalisation group map we specify conditions under which the non perturbative coordinate is contractive under a single renormalisation group step our framework is essentially combinatorial but its implementation relies on analytic results developed earlier in the series of papers the results of this paper are applied elsewhere to analyse the critical behaviour of the 4 dimensional continuous time weakly self avoiding walk and of the 4 dimensional n component varphi 4 model in particular the existence of a logarithmic correction to mean field scaling for the susceptibility can be proved for both models together with other facts about critical exponents and critical behaviour

on a finite range decomposition of the resolvent of a fractional power of the laplacian ii the torus in previous papers m1 m2 m3 we proved the existence as well as regularity of a finite range decomposition for the resolvent g alpha x y m 2 delta alpha over 2 m 2 1 x y for 0 alpha 2 and all real m in the lattice bf z d for dimension d ge 2 in this paper which is a continuation of the previous one we extend those results by proving the existence as well as regularity of a finite range decomposition for the same resolvent but now on the lattice torus bf z d l n 1 bf z d for d ge 2 provided m neq 0 and 0 alpha 2 we also prove differentiability and uniform continuity properties with respect to the resolvent parameter m 2 here l is any odd positive integer and n ge 2 is any positive integer

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

a non gaussian fixed point for φ4 in 4 e dimensions we consider the 4 quantum field theory in four dimensions the gaussian part of the measure is modified to simulate 4 e dimensions where e is small and positive we give a renormalization group analysis for the infrared behavior of the resulting model we find that the gaussian fixed point is unstable but that there is a hyperbolic non gaussian fixed point a distance e away in a neighborhood of this fixed point we construct the stable manifold

construction of the two dimensional sine gordon model for β 8π we present a rigorous renormalization group construction of the two dimensional massless and massive quantum sine gordon models in finite volume for the range 0 8 we prove analyticity in the coupling constant which implies the convergence of perturbation theory the field correlation functions and their generating functional are analyzed and shown to have the short distance asymptotics of the free field theory in the massive case the bounds are uniform in volume and we also obtain uniform estimates on the long distance decay of correlations

the structure of fluctuations in stochastic homogenization four quantities are fundamental in homogenization of elliptic systems in divergence form and in its applications the field and the flux of the solution operator applied to a general deterministic right hand side and the field and the flux of the corrector homogenization is the study of the large scale properties of these objects in case of random coefficients these quantities fluctuate and their fluctuations are a priori unrelated depending on the law of the coefficient field and in particular on the decay of its correlations on large scales these fluctuations may display different scalings and different limiting laws if any in this contribution we identify another crucial intrinsic quantity motivated by h convergence which we refer to as the homogenization commutator and is related to variational quantities first considered by armstrong and smart in the simplified setting of the random conductance model we show what we believe to be a general principle namely that the homogenization commutator drives at leading order the fluctuations of each of the four other quantities in a strong norm in probability which is expressed in form of a suitable two scale expansion and reveals the pathwise structure of fluctuations in stochastic homogenization in addition we show that the rescaled homogenization commutator converges in law to a gaussian white noise and we analyze to which precision the covariance tensor that characterizes the latter can be extracted from the representative volume element method this collection of results constitutes a new theory of fluctuations in stochastic homogenization that holds in any dimension and yields optimal rates extensions to the non symmetric continuum setting are also discussed the details of which are postponed to forthcoming works

on the spectrum of semi classical witten laplacians and schrodinger operators in large dimension we investigate the low lying spectrum of witten laplacians on forms of arbitrary degree in the semi classical limit and uniformly in the space dimension we show that under suitable assumptions implying that the phase function has a unique local minimum one obtains a number of clusters of discrete eigenvalues at the bottom of the spectrum moreover we are able to count the number of eigenvalues in each cluster we apply our results to certain sequences of schrodinger operators having strictly convex potentials and show that some well known results of semi classical analysis hold also uniformly in the dimension

witten laplacian method for the decay of correlations the aim of this paper is to apply direct methods to the study of integrals that appear naturally in statistical mechanics and euclidean field theory we provide weighted estimates leading to the exponential decay of the two point correlation functions for certain classical convex unbounded models the methods involve the study of the solutions of the witten laplacian equations associated with the hamiltonian of the system

u cross gram matrices and their invertibility abstract the gram matrix is defined for bessel sequences by combining synthesis with subsequent analysis operators if different sequences are used and an operator u is inserted we reach so called u cross gram matrices this can be seen as reinterpretation of the matrix representation of operators using frames in this paper we investigate some necessary or sufficient conditions for schatten p class properties and the invertibility of u cross gram matrices in particular we show that under mild conditions the pseudo inverse of a u cross gram matrix can always be represented as a u cross gram matrix with dual frames of the given ones we link some properties of u cross gram matrices to approximate duals finally we state several stability results more precisely it is shown that the invertibility of u cross gram matrices is preserved under small perturbations

grad o perturbations of massless gaussian fields we investigate weak perturbations of the continuum massless gaussian measure by a class of approximately local analytic functionals and use our general results to give a new proof that the pressure of the dilute dipole gas is analytic in the activity

multiscale representation of generating and correlation functions for some models of statistical mechanics and quantum field theory we consider models of statistical mechanics and quantum field theory in the euclidean formulation which are treated using renormalization group methods and where the action is a small perturbation of a quadratic action we obtain multiscale formulas for the generating and correlation functions aftern renormalization group transformations which bring out the relation with thenth effective action we derive and compare the formulas for different rgs the formulas for correlation functions involve 1 two propagators which are determined by a sequence of approximate wave function renormalization constants and renormalization group operators associated with the decomposition into scales of the quadratic form and 2 field derivatives of the nth effective action for the case of the block field function rg the formulas are especially simple and for asymptotic free theories only the derivatives at zero field are needed the formulas have been previously used directly to obtain bounds on correlation functions using information obtained from the analysis of effective actions the simplicity can be traced to an orthogonality of scales property which follows from an implicit wavelet structure other commonly used rgs do not have the orthogonality of scales property

quantum diffusion of the random schrodinger evolution in the scaling limit ii the recollision diagrams we consider random schrodinger equations on mathbb r d for d 3 with a homogeneous anderson poisson type random potential denote by the coupling constant and t the solution with initial data 0 the space and time variables scale as x sim lambda 2 kappa 2 t sim lambda 2 kappa with 0 0 d we prove that in the limit 0 the expectation of the wigner distribution of t converges weakly to the solution of a heat equation in the space variable x for arbitrary l2 initial data the proof is based on a rigorous analysis of feynman diagrams in the companion paper 10 the analysis of the non repetition diagrams was presented in this paper we complete the proof by estimating the recollision diagrams and showing that the main terms i e the ladder diagrams with renormalized propagator converge to the heat equation

scaling limit for a class of gradient fields with nonconvex potentials we consider gradient fields x x d whose law takes the gibbs boltzmann form z 1 exp x y v v x where the sum runs over nearest neighbors we assume that the potential v admits the representation v log d exp 1 2 2 where is a positive measure with compact support in 0 hence the potential v is symmetric but nonconvex in general while for strictly convex v s the translation invariant ergodic gradient gibbs measures are completely characterized by their tilt a nonconvex potential as above may lead to several ergodic gradient gibbs measures with zero tilt still every ergodic zero tilt gradient gibbs measure for the potential v above scales to a gaussian free field

a simple method for finite range decomposition of quadratic forms and gaussian fields we present a simple method to decompose the green forms corresponding to a large class of interesting symmetric dirichlet forms into integrals over symmetric positive semi definite and finite range properly supported forms that are smoother than the original green form this result gives rise to multiscale decompositions of the associated gaussian free fields into sums of independent smoother gaussian fields with spatially localized correlations our method makes use of the finite propagation speed of the wave equation and chebyshev polynomials it improves several existing results and also gives simpler proofs

a characterisation of the gaussian free field we prove that a random distribution in two dimensions which is conformally invariant and satisfies a natural domain markov property is a multiple of the gaussian free field this result holds subject only to a fourth moment assumption

finite range decomposition for gaussian measures with improved regularity abstract we consider a family of gradient gaussian vector fields on the torus z l n z d adams koteck muller and independently bauerschmidt established the existence of a uniform finite range decomposition of the corresponding covariance operators i e the covariance can be written as a sum of covariance operators supported on increasing cubes with diameter l k we improve this result and show that the decay behaviour of the kernels in fourier space can be controlled then we show the regularity of the integration map that convolves functionals with the partial measures of the finite range decomposition in particular the new finite range decomposition avoids the loss of regularity which arises in the renormalisation group approach to anisotropic problems in statistical mechanics

four dimensional weakly self avoiding walk with contact self attraction we consider the critical behaviour of the continuous time weakly self avoiding walk with contact self attraction on mathbb z 4 for sufficiently small attraction we prove that the susceptibility and correlation length of order p for any p 0 have logarithmic corrections to mean field scaling and that the critical two point function is asymptotic to a multiple of x 2 this shows that small contact self attraction results in the same critical behaviour as no contact self attraction a collapse transition is predicted for larger self attraction the proof uses a supersymmetric representation of the two point function and is based on a rigorous renormalisation group method that has been used to prove the same results for the weakly self avoiding walk without self attraction

the diffusion limit for reversible jump processes on z d with ergodic random bond conductivities we consider a reversible jump process on d whose jump rates themselves are random we show mean square convergence of this process under diffusion scaling to a limiting brownian motion with a certain diffusion matrix characterizing effective conductivity

large deviations and concentration properties for ϕ interface models we consider the massless field with zero boundary conditions outside d n d d n n d a suitable subset of d i e the continuous spin gibbs measure n on d n with hamiltonian given by h x y x y 1 v x y and x 0 for x d n c the interaction v is taken to be strictly convex and with bounded second derivative this is a standard effective model for a d 1 dimensional interface represents the height of the interface over the base d n due to the choice of scaling of the base we scale the height with the same factor by setting n n we study various concentration and relaxation properties of the family of random surfaces n and of the induced family of gradient fields n n as the discretization step 1 n tends to zero n in particular we prove a large deviation principle for n and show that the corresponding rate function is given by d u x dx where is the surface tension of the model this is a multidimensional version of the sample path large deviation principle we use this result to study the concentration properties of n under the volume constraint i e the constraint that 1 n d x dn n x stays in a neighborhood of a fixed volume v 0 and the hard wall constraint i e n x 0 for all x this is therefore a model for a droplet of volume v lying above a hard wall we prove that under these constraints the field n of rescaled heights concentrates around the solution of a variational problem involving the surface tension as it would be predicted by the phenomenological theory of phase boundaries our principal result however asserts local relaxation properties of the gradient field n n to the corresponding extremal gibbs states thus our approach has little in common with traditional large deviation techniques and is closer in spirit to hydrodynamic limit type of arguments the proofs have both probabilistic and analytic aspects essential analytic tools are p estimates for elliptic equations and the theory of young measures on the side of probability tools a central role is played by the helffer sjostrand 31 pde representation for continuous spin systems which we rewrite in terms of random walk in random environment and by recent results of t funaki and h spohn 25 on the structure of gradient fields

difference analogue of the lemma on the logarithmic derivative with applications to difference equations the lemma on the logarithmic derivative of a meromorphic function has many applications in the study of meromorphic functions and ordinary dierential equa tions in this paper a dierence analogue of the logarithmic derivative lemma is presented and then applied to prove a number of results on meromorphic solutions of complex dierence equations these results include a dierence analogue of the clunie lemma as well as other results on the value distribution of solutions

finite range decomposition of gaussian processes let be the finite difference laplacian associated to the lattice z d for dimension d 3 a 0 and l a sufficiently large positive dyadic integer we prove that the integral kernel of the resolvent g a a 1 can be decomposed as an infinite sum of positive semi definite functions v n of finite range v n x y 0 for x y o l n equivalently the gaussian process on the lattice with covariance g a admits a decomposition into independent gaussian processes with finite range covariances for a 0 v n has a limiting scaling form l n left d 2 right gamma c left tfrac x y l n right as n as a corollary such decompositions also exist for fractional powers 2 0 2 the results of this paper give an alternative to the block spin renormalization group on the lattice

quantum field theory in ninety minutes on presente divers aspects de la theorie quantique des champs la theorie p 2 a 2 dimensions la theorie 3 4 a 3 dimensions la theorie de yang mills a 4 dimensions

erratum to on a finite range decomposition of the resolvent of a fractional power of the laplacian erratum to on a finite range decomposition of the resolvent of a fractional power of the laplacian

critical φ 4 3 e the euclidean 4 3 e model in r3 corresponds to a perturbation by a 4 interaction of a gaussian measure on scalar fields with a covariance depending on a real parameter e in the range 0 e 1 for e 1 one recovers the covariance of a massless scalar field in r3 for e 0 4 is a marginal interaction for 0 e 0 sufficiently small there exists a non gaussian fixed point with one unstable direction of the renormalization group iterations we construct the stable critical manifold near this fixed point and prove that under renormalization group iterations the critical theories converge to the fixed point

annealed estimates on the green function we consider a random uniformly elliptic coefficient field a x on the d dimensional integer lattice mathbb z d we are interested in the spatial decay of the quenched elliptic green function g a x y next to stationarity we assume that the spatial correlation of the coefficient field decays sufficiently fast to the effect that a logarithmic sobolev inequality holds for the ensemble langle cdot rangle we prove that all stochastic moments of the first and second mixed derivatives of the green function that is langle nabla x g x y p rangle and langle nabla x nabla y g x y p rangle have the same decay rates in x y gg 1 as for the constant coefficient green function respectively this result relies on and substantially extends the one by delmotte and deuschel probab theory relat fields 133 358 390 2005 which optimally controls second moments for the first derivatives and first moments of the second mixed derivatives of g that is langle nabla x g x y 2 rangle and langle nabla x nabla y g x y rangle as an application we are able to obtain optimal estimates on the random part of the homogenization error even for large ellipticity contrast

basic definition and properties of bessel multipliers this paper introduces the concept of bessel multipliers these operators are defined by a fixed multiplication pattern which is inserted between the analysis and synthesis operators the proposed concept unifies the approach used for gabor multipliers for arbitrary analysis synthesis systems which form bessel sequences like wavelet or irregular gabor frames the basic properties of this class of operators are investigated in particular the implications of summability properties of the symbol for the membership of the corresponding operators in certain operator classes are specified as a special case the multipliers for riesz bases are examined and it is shown that multipliers in this case can be easily composed and inverted finally the continuous dependence of a bessel multiplier on the parameters i e the involved sequences and the symbol in use is verified using a special measure of similarity of sequences

the renormalization group according to balaban ii large fields this is an expository account of balaban s approach to the renormalization group the method is illustrated with a treatment of the ultraviolet problem for the scalar 4 model on a toroidal lattice in dimension d 3 in this paper we control the large field contribution to the partition function

renormalization group flow of a hierarchical sine gordon model by partial differential equations we use a renormalization group differential equation to rigorously control the renormalization group flow in a hierarchical lattice sine gordon field theory in the kosterlitz thouless phase

renormalization group approach to interacting polymerised manifolds we propose to study the infrared behaviour of polymerised or tethered random manifolds of dimension d interacting via an exclusion condition with a fixed impurity in d dimensional euclidean space in which the manifold is embedded in this paper we take d 1 but modify the underlying free gaussian covariance thereby changing the canonical scaling dimension of the gaussian random field so as to simulate a polymerised manifold with fractional dimension the canonical dimension of the coupling constant is where 2 is the canonical scaling dimension of the gaussian embedding field is held strictly positive and sufficiently small for 0 sufficiently small we prove for this model that the iterations of wilson s renormalisation group transformations converge to a non gaussian fixed point although is small our analysis is non perturbative in a similar model was studied earlier cm in the hierarchical approximation

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

multiscale formalism for correlation functions of fermions infrared analysis of the tridimensional gross neveu model we present a multiscale formalism for fermionic systems with a smooth uv cutoff establishing a trivial link between the correlation functions and the effective potential flow and study the k point truncated functions of the tridimensional gross neveu model a new efficient method is used to bound these correlation functions and show polynomial tree decay for long distances we are guided by a block lattice mechanism with a property of orthogonality between terms in different scales which leads to simple formulas for the correlations

on the convergence to the continuum of finite range lattice covariances in j stat phys 115 415 449 2004 brydges guadagni and mitter proved the existence of multiscale expansions of a class of lattice green s functions as sums of positive definite finite range functions called fluctuation covariances the lattice green s functions in the class considered are integral kernels of inverses of second order positive self adjoint elliptic operators with constant coefficients and fractional powers thereof the rescaled fluctuation covariance in the nth term of the expansion lives on a lattice with spacing l n and satisfies uniform bounds our main result in this note is that the sequence of these terms converges in appropriate norms at a rate l n 2 to a smooth positive definite finite range continuum function

decay of correlations and uniqueness of gibbs lattice systems with nonquadratic interaction the aim of this paper is to develop the classical lattice models with unbounded spin to the case of nonquadratic polynomial interaction we demonstrate that the distinct relation between the growths of potentials leads to the uniqueness and the fast decay of correlations for gibbs measure

frames and semi frames loosely speaking a semi frame is a generalized frame for which one of the frame bounds is absent more precisely given a total sequence in a hilbert space we speak of an upper resp lower semi frame if only the upper resp lower frame bound is valid equivalently for an upper semi frame the frame operator is bounded but has an unbounded inverse whereas a lower semi frame has an unbounded frame operator with bounded inverse we study mostly upper semi frames both in the continuous case and in the discrete case and give some remarks for the dual situation in particular we show that reconstruction is still possible in certain cases

on the averaged green s function of an elliptic equation with random coefficients we consider a divergence form elliptic difference operator on the lattice mathbb z d with a coefficient matrix that is an i i d perturbation of the identity matrix recently bourgain introduced novel techniques from harmonic analysis to prove the convergence of the feshbach schur perturbation series related to the averaged green s function of this model our main contribution is a refinement of bourgain s approach which improves the key decay rate from 2d epsilon to 3d epsilon the optimal decay rate is conjectured to be 3d as an application we derive estimates on higher derivatives of the averaged green s function which go beyond the second derivatives considered by delmotte deuschel and related works

quantum diffusion of the random schrodinger evolution in the scaling limit we consider random schrodinger equations on r d for d 3 with a homogeneous anderson poisson type random potential denote by the coupling constant and psi t the solution with initial data psi 0 the space and time variables scale as x sim lambda 2 varkappa 2 text and t sim lambda 2 varkappa text with 0 varkappa varkappa 0 left d right we prove that in the limit 0 the expectation of the wigner distribution of psi t converges weakly to the solution of a heat equation in the space variable x for arbitrary l 2 initial data

next order asymptotic expansion for n marginal optimal transport with coulomb and riesz costs abstract motivated by a problem arising from density functional theory we provide the sharp next order asymptotics for a class of multimarginal optimal transport problems with cost given by singular long range pairwise interaction potentials more precisely we consider an n marginal optimal transport problem with n equal marginals supported on r d and with cost of the form i j x i x j s in this setting we determine the second order term in the n asymptotic expansion of the minimum energy for the long range interactions corresponding to all exponents 0 s d we also prove a small oscillations property for this second order energy term our results can be extended to a larger class of models than power law type radial costs such as non rotationally invariant costs the key ingredient and main novelty in our proofs is a robust extension and simplification of the fefferman gregg decomposition 20 26 extended here to our class of kernels and which provides a unified method valid across our full range of exponents our first result generalizes a recent work of lewin lieb and seiringer 36 who dealt with the second order term for the coulomb case s 1 d 3

a complete renormalization group trajectory between two fixed points we give a rigorous nonperturbative construction of a massless discrete trajectory for wilson s exact renormalization group the model is a three dimensional euclidean field theory with a modified free propagator the trajectory realizes the mean field to critical crossover from the ultraviolet gaussian fixed point to an analog recently constructed by brydges mitter and scoppola of the wilson fisher nontrivial fixed point

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