accessory parameters in conformal mapping exploiting the isomonodromic tau function for painleve vi we present a novel method to solve the accessory parameter problem arising in constructing conformal maps from a canonical simply connected planar region to the interior of a circular arc quadrilateral the schwarz christoffel accessory parameter problem relevant when all sides have zero curvature is also captured within our approach the method exploits the isomonodromic tau function associated with the painleve vi equation recently these tau functions have been shown to be related to certain correlation functions in conformal field theory and asymptotic expansions have been given in terms of tuples of the young diagrams after showing how to extract the monodromy data associated with the target domain we show how a numerical approach based on the known asymptotic expansions can be used to solve the conformal mapping accessory parameter problem the viability of this new method is demonstrated by explicit examples and we discuss its extension to circular arc polygons with more than four sides

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

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

numerical conformal mapping of circular arc polygons abstract howell l h numerical conformal mapping of circular arc polygons journal of computational and applied mathematics 46 1993 7 28 solutions to the schwarzian differential equation are conformal maps from the upper half plane to circular arc polygons plane regions bounded by straight line segments and arbitrary arcs of circles we develop methods for numerically integrating this equation both directly and through the use of a related linear differential equation particular attention is given to the behavior near corner singularities we also derive alternate versions of the transformation which map from the unit disk and from an infinite strip while the former may be of primarily theoretical interest the latter can be used to map highly elongated regions such as channels for internal flow problems such regions are difficult or impossible to map from the disk or the half plane due to the so called crowding phenomenon

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

an analogy of bol s result on jacobi forms and siegel modular forms in this paper the analogy of bol s result to the several variable function case is discussed one shows how to construct siegel modular forms and jacobi forms of higher degree respectively using bol s result

block renormalization group for euclidean fermions block renormalization group transformations rgt for lattice and continuum euclidean fermions in d dimensions are developed using fermionic integrals with exponential and function weight functions for the free field the sequence of actionsdk generated by the rgt from d the dirac operator are shown to have exponential decay uniform ink after rescaling to the unit lattice it is shown that the two point functiond 1 admits a simple telescopic sum decomposition into fluctuation two point functions which for the exponential weight rgt have exponential decay contrary to rg intuition the sequence of rescaled actions corresponding to the function rgt do not have uniform exponential decay and we give examples of initial actions in one dimension where this phenomena occurs for the exponenential weight rgt also

numerical conformal mapping using cross ratios and delaunay triangulation we propose a new algorithm for computing the riemann mapping of the unit disk to a polygon also known as the schwarz christoffel transformation the new algorithm crdt for cross ratios of the delaunay triangulation is based on cross ratios of the prevertices and also on cross ratios of quadrilaterals in a delaunay triangulation of the polygon the crdt algorithm produces an accurate representation of the riemann mapping even in the presence of arbitrary long thin regions in the polygon unlike any previous conformal mapping algorithm we believe that crdt solves all difficulties with crowding and global convergence although these facts depend on conjectures that we have so far not been able to prove we demonstrate convergence with computational experiments the riemann mapping has applications in two dimensional potential theory and mesh generation we demonstrate crdt on problems in long thin regions in which no other known algorithm can perform comparably

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

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

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

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

regularity and decay of lattice green s functions in the paper we study a class of lattice covariant laplace operators with external gauge fields we prove that these operators are positive and that their green s functions decay exponentially they also have regularity properties similar to continuous space green s functions all the bounds are uniform in the lattice spacing

multilinear operators on siegel modular forms of genus 1 and 2 classically there are many interesting connections between differential operators and the theory of elliptic modular forms and many interesting results have been explored in particular it has been known for some time how to obtain an elliptic modular form from the derivatives ofn elliptic modular forms which has already been studied in detail by r rankin in 9 and 10 when n 2 as a special case of rankin s result in 10 h cohen constructed certain covariant bilinear operators which he used to obtain modular forms with interesting fourier coefficients 4 later these covariant bilinear operators were called rankin cohen operators by d zagier who studied their algebraic relations 12 13 as well as connections with formal pseudodifferential operators 5 furthermore rankin cohen operators are shown to appear as the various terms in the convergent expansion of the composition of two symbols in a certain symbolic calculus associated with sl 2 r 11 recently the rankin cohen type bracket operator on jacobi forms and siegel modular forms has been studied using the heat operator and differential operator respectively in fact it was shown how to construct explicitly jacobi forms and siegel modular forms of genus 2 with the rankin cohen type bracket operators involving the heat operator and the determinant differential operators respectively 1

renormalization group approach to lattice gauge field theories ii cluster expansions the fluctuation field integral constructed in part i is represented by the exponentiated cluster expansion it is proved that the terms of the expansion satisfy the inductive assumptions this completes the construction of the sequence of effective actions in the small field approximation

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

averaging operations for lattice gauge theories usually renormalization group transformations are defined by some averaging operations in this paper we study such operations for lattice gauge fields and for gauge transformations we are interested especially in characterizing some classes of field configurations on which the averaging operations are regular e g analytic these results will be used in subsequent papers on the renormalization group method in lattice gauge theories

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

lattice and continuum wavelets and the block renormalization group the authors obtain a resolution of the identity operator for functions on a lattice var epsilon z sup d which is derived from the block renormalization group the authors use eigenfunctions of the terms of the decomposition to form a basis for l sub 2 var epsilon z sup d and show how the basis is generated from lattice wavelets the lattice spacing var epsilon is taken to zero and continuum wavelets are obtained 12 refs

convergence of theu 1 4 lattice gauge theory to its continuum limit it is shown that in four space time dimensions the compactu 1 lattice gauge theory with general energy function converges to a renormalized free electromagnetic field on the current sector as the lattice spacing approaches zero provided the coupling constant is sufficiently large for the wilson energy function it is possible by judicious choice of the gibbs state to get convergence for arbitrary coupling strengths furthermore for all but a countable number of values of the coupling constant the limit exists and is independent of the particular state chosen to define the lattice model

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

a modified schwarz christoffel mapping for regions with piecewise smooth boundaries a method where polygon corners in schwarz christoffel mappings are rounded is used to construct mappings from the upper half plane to regions bounded by arbitrary piecewise smooth curves from a given curve a polygon is constructed by taking tangents to the curve in a number of carefully chosen so called tangent points the schwarz christoffel mapping for that polygon is then constructed and modified to round the corners since such a modification causes effects on the polygon outside the rounded corners the parameters in the mapping have to be redetermined this is done by comparing side lengths in tangent polygons to the given curve and the curve produced by the modified schwarz christoffel mapping the set of equations that this comparison gives can normally be solved using a quasi newton method the resulting function maps the upper half plane on a region bounded by a curve that apart from possible vertices is smooth i e one time continuously differentiable that passes through the tangent points on the given curve has the same direction as the given curve in these points and changes direction monotonically between them furthermore where the original curve has a vertex the constructed curve has a vertex with the same inner angle the method is especially useful for unbounded regions with smooth boundary curves that pass infinity as straight lines such as channels with parallel walls at the ends these properties are kept in the region produced by the constructed mapping

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

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

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

conformal mappings to a doubly connected polycircular arc domain the explicit construction of the conformal mapping of a concentric annulus to a doubly connected polygonal domain was first reported by akhiezer in 1928 the construction of an analogous formula for the case of a polycircular arc domain i e for a doubly connected domain whose boundaries are a union of circular arc segments has remained an important open problem in this paper we present this explicit formula we first introduce a new method for deriving the classical formula of akhiezer and then show how to generalize the method to the case of a doubly connected polycircular arc domain as an analytical check of the formula a special exact solution for a doubly connected polycircular arc mapping is derived and compared with that obtained from the more general construction as an illustrative example a doubly connected polycircular arc domain arising in a classic potential flow problem considered in the last century by lord rayleigh is considered in detail

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

acoustic transmission in curved ducts with varying cross sections methods for predicting the acoustic propagation properties of ducts are developed and analysed for ducts that are asymptotically straight with constant crosssection with application to splitter si

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 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

schwarz christoffel mappings a general approach abstract a general method is developed for constructing equations of the schwarz christoffel type these equations define conformal transformations between regions of the complex plane each of whose boundaries may consist of straight or continuously curved line segments and corners the more familiar types of schwarz christoffel equations are rederived and some new types are obtained

modified schwarz christoffel mappings using approximate curve factors the schwarz christoffel mapping from the upper half plane to a polygonal region in the complex plane is an integral of a product with several factors where each factor corresponds to a certain vertex in the polygon different modifications of the schwarz christoffel mapping in which factors are replaced with the so called curve factors to achieve polygons with rounded corners are known since long times among other requisites the arguments of a curve factor and its correspondent scl factor must be equal outside some closed interval on the real axis in this paper the term approximate curve factor is defined such that many of the already known curve factors are included as special cases additionally by alleviating the requisite on the argument from exact to asymptotic equality new types of curve factors are introduced while traditional curve factors have a c 1 regularity c regular approximate curve factors can be constructed resulting in smooth boundary curves when used in conformal mappings applications include modelling of wave scattering in waveguides when using approximate curve factors in modified schwarz christoffel mappings numerical conformal mappings can be constructed that preserve two important properties in the waveguides first the direction of the boundary curve can be well controlled especially towards infinity where the application requires two straight parallel walls second a smooth c boundary curve can be achieved

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

renormalization group approach to lattice gauge field theories i generation of effective actions in a small field approximation and a coupling constant renormalization in four dimensions on etudie des theories de champs de jauge pures a 4 dimensions par l approche du groupe de renormalisation on se restreint a l approximation de champ petit dans cette region on construit une suite d actions effectives localisees par des developpements en amas dans des transformations de renormalisation a un pas

a low temperature expansion for classical n vector models ii renormalization group equations this paper continues the analysis of the low temperature expansions for classicaln vector models started in 1 a main part of it is a derivation of renormalization group equations and a construction of their solutions to do this we have to introduce a fluctuation integral connected with a next renormalization transformation and to make its preliminary analysis the results of the paper are summarized in theorems stating that the renormalization transformation preserves the space of densitites or actions described inductively in 1

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