witten s invariants of rational homology spheres at prime values of k and trivial connection contribution we establish a relation between the coefficients of asymptotic expansion of the trivial connection contribution to witten s invariant of rational homology spheres and the invariants that t ohtsuki extracted from witten s invariant at prime values ofk we also rederive the properties of primek invariants discovered by h murakami and t ohtsuki we do this by using the bounds on taylor series expansion of the jones polynomial of algebraically split links studied in our previous paper these bounds are enough to prove that ohtsuki s invariants are of finite type the relation between ohtsuki s invariants and trivial connection contribution is verified explicitly for lens spaces and seifert manifolds

on cohen macaulayness of algebras generated by generalized power sums generalized power sums are linear combinations of ith powers of coordinates we consider subalgebras of the polynomial algebra generated by generalized power sums and study when such algebras are cohen macaulay it turns out that the cohen macaulay property of such algebras is rare and tends to be related to quantum integrability and representation theory of cherednik algebras using representation theoretic results and deformation theory we establish cohen macaulayness of the algebra of q t deformed power sums defined by sergeev and veselov and of some generalizations of this algebra proving a conjecture of brookner corwin etingof and sam we also apply representation theoretic techniques to studying m quasi invariants of deformed calogero moser systems in an appendix to this paper m feigin uses representation theory of cherednik algebras to compute hilbert series for such quasi invariants and show that in the case of one light particle the ring of quasi invariants is gorenstein

a variational formula for the free energy of an interacting many particle system we consider n bosons in a box in r d with volume n rho under the influence of a mutually repellent pair potential the particle density p is an element of 0 infinity is kept fixed our main result is the identification of the limiting free energy f beta p at positive temperature 1 beta in terms of an explicit variational formula for any fixed rho if beta is sufficiently small and for any fixed beta if rho is sufficiently small the thermodynamic equilibrium is described by the symmetrized trace of e beta h n where h n denotes the corresponding hamilton operator the well known feynman kac formula reformulates this trace in terms of n interacting brownian bridges due to the symmetrization the bridges are organized in an ensemble of cycles of various lengths the novelty of our approach is a description in terms of a marked poisson point process whose marks are the cycles this allows for an asymptotic analysis of the system via a large deviations analysis of the stationary empirical field the resulting variational formula ranges over random shift invariant marked point fields and optimizes the sum of the interaction and the relative entropy with respect to the reference process in our proof of the lower bound for the free energy we drop all interaction involving infinitely long cycles and their possible presence is signalled by a loss of mass of the finitely long cycles in the variational formula in the proof of the upper bound we only keep the mass on the finitely long cycles we expect that the precise relationship between these two bounds lies at the heart of bose einstein condensation and intend to analyze it further in future

generalized covariations local time and stratonovich ito s formula for fractional brownian motion with hurst index h 1 4 given a locally bounded real function g we examine the existence of a 4 covariation g b h b h b h b h where b h is a fractional brownian motion with a hurst index h ge tfrac 1 4 we provide two essential applications first we relate the 4 covariation to one expression involving the derivative of local time in the case h tfrac 1 4 generalizing an identity of bouleau yor type well known for the classical brownian motion a second application is an ito formula of stratonovich type for f b h the main difficulty comes from the fact b h has only a finite 4 variation

forward backward and symmetric stochastic integration we define three types of non causal stochastic integrals forward backward and symmetric our approach consists in approximating the integrator two optics are considered the first one is based on traditional usual stochastic calculus and the second one on wiener distributions

a countable representation of the fleming viot measure valued diffusion the fleming viot measure valued diffusion arises as the infinite population limit of various discrete genetic models with general type space the paper gives a countable construction of the process as the empirical measure carried by a certain interactive particle system this explicit representation facilitates the study of various properties of the fleming viot process the construction also carries versions of the familiar genealogical processes from population genetics in particular kingman s coalescent thus unifying the genealogical and measure valued approaches to the subject

the equivalence of ensembles for classical systems of particles for systems of particles in classical phase space with standard hamiltonian we consider spatially averaged microcanonical gibbs distributions in finite boxes we show that infinite volume limits along suitable subsequences exist and are grand canonical gibbs measures on the way we establish a variational formula for the thermodynamic entropy density as well as a variational characterization of grand canonical gibbs measures

quenched asymptotics of the ground state energy of random schrodinger operators with scaled gibbsian potentials this article describes the almost sure infinite volume asymptotics of the ground state energy of random schrodinger operators with scaled gibbsian potentials the random potential is obtained by distributing soft obstacles according to an infinite volume grand canonical tempered gibbs measure with a superstable pair interaction there is no restriction on the strength of the pair interaction it may be taken e g at a critical point the potential is scaled with the box size in a critical way i e the scale is determined by the typical size of large deviations in the gibbsian cloud the almost sure infinite volume asymptotics of the ground state energy is described in terms of two equivalent deterministic variational principles involving only thermodynamic quantities the qualitative behaviour of the ground state energy asymptotics is analysed depending on the dimension and on the holder exponents of the free energy density it is identified which cases lead to a phase transition of the asymptotic behaviour of the ground state energy

existence of gibbsian point processes with geometry dependent interactions we establish the existence of stationary gibbsian point processes for interactions that act on hyperedges between the points for example such interactions can depend on delaunay edges or triangles cliques of voronoi cells or clusters of k nearest neighbors the classical case of pair interactions is also included the basic tools are an entropy bound and stationarity

explicit computation of the drinfeld associator in the case of the fundamental representation of gl n we solve the regularized knizhnik zamolodchikov equation and find an explicit expression for the drinfeld associator we restrict ourselves to the case of the fundamental representation of gl n several tests of the results are presented it can be explicitly seen that components of this solution for the associator coincide with certain components of the wzw conformal block for primary fields we introduce the symmetrized version of the drinfeld associator by dropping the odd terms the symmetrized associator gives the same knot invariants but has a simpler structure and is fully characterized by one symmetric function which we call the drinfeld prepotential

hydrodynamical limit for a hamiltonian system with weak noise starting from a general hamiltonian system with superstable pairwise potential we construct a stochastic dynamics by adding a noise term which exchanges the momenta of nearby particles we prolve that in the scaling limit the time conserved quantities energy momenta and density satisfy the euler equation of conservation laws up to a fixed timet provided that the euler equation has a smooth solution with a given initial data up to timet the strength of the noise term is chosen to be very small but nonvanishing so that it disappears in the scaling limit

r local delaunay inhibition model unlike in the classical framework of gibbs point processes usually acting on the complete graph in the context of nearest neighbour gibbs point processes the nonnegativeness of the interaction functions do not ensure the local stability property this paper introduces domain wise but not pointwise inhibition stationary gibbs models based on some tailor made delaunay subgraphs all of them are subgraphs of the r local delaunay graph defined as the delaunay subgraph specifically not containing the edges of delaunay triangles with circumscribed circles of radii greater than some large positive real value r the usual relative compactness criterion for point processes needed for the existence result is directly derived from the ruelle bound of the correlation functions furthermore assuming only the nonnegativeness of the energy function we have managed to prove the existence of the existence of r local delaunay stationary gibbs states based on nonnegative interaction functions thanks to the use of the compactness of sublevel sets of the relative entropy

complete equivalence of the gibbs ensembles for one dimensional markov systems we show the equivalence of the gibbs ensembles at the level of measures for one dimensional markov systems with arbitrary boundary conditions that is the limit of the microcanonical gibbs ensemble is a gibbs measure with an interaction depending on the microcanonical constraint in fact the usual microcanonical condition is replaced by the sharper constraint that all type frequencies of neighboring spins including the boundary spins are fixed when conditioning on a set of different frequencies of neighboring spins compatible with physical quantities like energy density we get the usual microcanonical ensemble we show that the limit is a gibbs measure for a nearest neighbor potential depending on the pair measure which maximizes the entropy on the given set of pair measures for this we show the large deviation property of the pair empirical measure for arbitrary boundary conditions we establish analogous results for finite range potentials

joint convergence along different subsequences of the signed cubic variation of fractional brownian motion the purpose of this paper is to study the convergence in distribution of two subsequences of the signed cubic variation of the fractional brownian motion with hurst parameter h 1 6 we prove that under some conditions on both subsequences the limit is a two dimensional brownian motion whose components may be correlated and we find explicit formulae for its covariance function

a change of variable formula with ito correction term we consider the solution u x t to a stochastic heat equation for fixed x the process f t u x t has a nontrivial quartic variation it follows that f is not a semimartingale so a stochastic integral with respect to f cannot be defined in the classical ito sense we show that for sufficiently differentiable functions g x t a stochastic integral g f t t d f t exists as a limit of discrete midpoint style riemann sums where the limit is taken in distribution in the skorokhod space of cadlag functions moreover we show that this integral satisfies a change of variable formula with a correction term that is an ordinary ito integral with respect to a brownian motion that is independent of f

a change of variable formula for the 2d fractional brownian motion of hurst index bigger or equal to 1 4 we prove a change of variable formula for the 2d fractional brownian motion of index h bigger of equal to 1 4 for h strictly bigger than 1 4 our formula coincides with that obtained by using the rough paths theory for h 1 4 the more interesting case there is an additional term that is a classical wiener integral against an independent standard brownian motion

large deviations and the maximum entropy principle for marked point random fields we establish large deviation principles for the stationary and the individual empirical fields of poisson and certain interacting random fields of marked point particles in d the underlying topologies are induced by a class of not necessarily bounded local functions and thus finer than the usual weak topologies our methods yield further that the limiting behaviour of conditional poisson distributions as well as certain distributions of gibbsian type is governed by the maximum entropy principle we also discuss various applications and examples

a ribbon hopf algebra approach to the irreducible representations of centralizer algebras the brauer birman wenzl and type a iwahori hecke algebras abstract we show how the ribbon hopf algebra structure on the drinfel d jimbo quantum groups of types a b c and d can be used to derive formulas giving explicit realizations of the irreducible representations of the iwahori hecke algebras of type a and the birman wenzl algebras we use this derivation to give explicit realizations of the irreducible representations of the brauer algebras as well the derivation is accomplished by way of a combination of techniques from operator algebras quantum groups and the theory of 3 manifold invariants although our applications are in the cases of the quantum groups of types a b c and d most of the aspects of our approach apply in the general setting of ribbon hopf algebras

large deviations and the equivalence of ensembles for gibbsian particle systems with superstable interaction for gibbsian systems of particles inr d we investigate large deviations of the translation invariant empirical fields in increasing boxes the particle interaction is given by a superstable regular pair potential the large deviation principle is established for systems with free or periodic boundary conditions and under a stronger stability hypothesis on the potential for systems with tempered boundary conditions and for tempered infinite volume gibbs measures as a by product we obtain the gibbs variational formula for the pressure we also prove the asymptotic equivalence of microcanonical and grand canonical gibbs distributions and establish a variational expression for the thermodynamic entropy density

positivity of the self diffusion matrix of interacting brownian particles with hard core we prove the positivity of the self diffusion matrix of interacting brownian particles with hard core when the dimension of the space is greater than or equal to 2 here the self diffusion matrix is a coefficient matrix of the diffusive limit of a tagged particle we will do this for all activities z 0 of gibbs measures in particular for large z the case of high density particles a typical example of such a particle system is an infinite amount of hard core brownian balls

phase transitions for dilute particle systems with lennard jones potential we consider a classical dilute particle system in a large box with pair interaction given by a lennard jones type potential the inverse temperature is picked proportionally to the logarithm of the particle density we identify the free energy per particle in terms of a variational formula and show that this formula exhibits a cascade of phase transitions as the temperature parameter ranges from zero to infinity loosely speaking the particle system separates into spatially distant components in such a way that within each phase all components are of the same size which is the larger the lower the temperature the main tool in our proof is a new large deviation principle for sparse point configurations

variations of the solution to a stochastic heat equation we consider the solution to a stochastic heat equation this solution is a random function of time and space for a fixed point in space the resulting random function of time f t has a nontrivial quartic variation this process therefore has infinite quadratic variation and is not a semimartin gale it follows that the classical ito calculus does not apply motivated by heuristic ideas about a possible new calculus for this process we are led to study modifications of the quadratic variation namely we modify each term in the sum of the squares of the increments so that it has mean zero we then show that these sums as functions of t converge weakly to brownian motion

existence of delaunay pairwise gibbs point process with superstable component the present stuffy deals with the existence of delaunay pairwise gibbs point process with superstable component by using the well known preston theorem in particular we prove the stability the lower regularity and the quasilocality properties of the delaunay model

witten reshetikhin turaev invariants of seifert manifolds for seifert homology spheres we derive a holomorphic function of k whose value at integer k is the sl2 witten reshetikhin turaev invariant zk at q exp 2 i k this function is expressed as a sum of terms which can be naturally corresponded to the contributions of flat connections in the stationary phase expansion of the witten chern simons path integral the trivial connection contribution is found to have an asymptotic expansion in powers of k 1 which for k an odd prime power converges k adically to the exact total value of the invariant zk at that root of unity evaluations at rational k are also discussed using similar techniques an expression for the coloured jones polynomial of a torus knot is obtained providing a trivial connection contribution which is an analytic function of the colour this demonstrates that the stationary phase expansion of the chern simons witten theory is exact for seifert manifolds and for torus knots in s3 the possibility of generalising such results is also discussed

string theory and the kauffman polynomial we propose a new precise integrality conjecture for the colored kauffman polynomial of knots and links inspired by large n dualities and the structure of topological string theory on orientifolds according to this conjecture the natural knot invariant in an unoriented theory involves both the colored kauffman polynomial and the colored homfly polynomial for composite representations i e it involves the full homfly skein of the annulus the conjecture sheds new light on the relationship between the kauffman and the homfly polynomials and it implies for example rudolph s theorem we provide various non trivial tests of the conjecture and we sketch the string theory arguments that lead to it

polynomial invariants for torus knots and topological strings we make a precision test of a recently proposed conjecture relating chern simons gauge theory to topological string theory on the resolution of the conifold first we develop a systematic procedure to extract string amplitudes from vacuum expectation values vevs of wilson loops in chern simons gauge theory and then we evaluate these vevs in arbitrary irreducible representations of su n for torus knots we find complete agreement with the predictions derived from the target space interpretation of the string amplitudes we also show that the structure of the free energy of topological open string theory gives further constraints on the chern simons vevs our work provides strong evidence towards an interpretation of knot polynomial invariants as generating functions associated to enumerative problems

large deviations and maximum entropy principle for interacting random fields on mathbb z d we present a new approach to the principle of large deviations for the empirical field of a gibbsian random field on the integer lattice zd this approach has two main features first we can replace the traditional weak topology by the finer topology of convergence of cylinder probabilities and thus obtain estimates which are finer and more widely applicable second we obtain as an immediate consequence a limit theorem for conditional distributions under conditions on the empirical field the limits being those predicted by the maximum entropy principle this result implies a general version of the equivalence of gibbs ensembles stating that every microcanonical limiting state is a grand canonical equilibrium state we also prove a converse to the last statement and discuss some applications 0 introduction as is well known the study of the asymptotic probabilities of large fluctuations of time averages or space averages away from the mean is based on two fundamental principles the principle of large deviations and the maximum entropy principle the former provides the exact rate of exponential decay of the fluctuation probabilities whereas the latter predicts the limiting conditional distribution under the condition that the fluctuations are large it is obvious that these principles are intimately related in this paper we investigate these principles in the case of interacting random fields on the integer lattice 7zd the setup is the following first we let e 4 be any measurable space we shall assume throughout that e a is standard borel but we shall avoid making any explicit topological assumptions on e so we do not assume that e is polish next we let s zd be the d dimensional integer lattice and q f e gas the associated product space in lattice models of statistical mechanics e is called the state space or single spin space and q f the configuration space we let j s f denote the set of all probability

on the virasoro constraints for torus knots we construct a virasoro algebra of differential operators for the matrix model for torus knots these operators generate various relations between wilson loops then we discuss the operators constructed and corresponding relations in the stability limit we also give a series of examples

asymptotic behavior of weighted quadratic variations of fractional brownian motion the critical case h 1 4 we derive the asymptotic behavior of weighted quadratic variations of fractional brow nian motion b with hurst index h 1 4 this completes the only missing case in a very recent work by i nourdin d nualart and c a tudor moreover as an application we solve a recent conjecture of k burdzy and j swanson on the asymptotic behavior of the riemann sums with alternating signs associated to b

schur polynomials and biorthogonal random matrix ensembles the study of the average of schur polynomials over a stieltjes wigert ensemble has been carried out by dolivet and tierz j math phys 48 023507 2007 e print arxiv hep th 0609167 where it was shown that it is equal to quantum dimensions using the same approach we extend the result to the biorthogonal case we also study using the littlewood richardson rule some particular cases of the quantum dimension result finally we show that the notion of giambelli compatibility of schur averages introduced by borodin et al adv appl math 37 209 2006 e print arxiv math ph 0505021 also holds in the biorthogonal setting

phase transition in the nearest neighbor continuum potts model in the present study we establish a phase transition in the nearest neighbor continuum potts model the repulsion between particles of different type acts only on a nearest neighbor graph more precisely a subgraph of the delaunay graph this work is an adaptation of the lebowitz and lieb soft core continuum potts model

representations of rational cherednik algebras with minimal support and torus knots abstract in this paper we obtain several results about representations of rational cherednik algebras and discuss their applications our first result is the cohen macaulayness property as modules over the polynomial ring of cherednik algebra modules with minimal support our second result is an explicit formula for the character of an irreducible minimal support module in type a n 1 for c m n and an expression of its quasispherical part i e the isotypic part of hooks in terms of the homfly polynomial of a torus knot colored by a young diagram we use this formula and the work of calaque enriquez and etingof to give explicit formulas for the characters of the irreducible equivariant d modules on the nilpotent cone for sl m our third result is the construction of the koszul bgg complex for the rational cherednik algebra which generalizes the construction of the koszul bgg resolution from 3 and 21 and the calculation of its homology in type a we also show in type a that the differentials in the koszul bgg complex are uniquely determined by the condition that they are nonzero homomorphisms of modules over the cherednik algebra finally our fourth result is the symmetry theorem which identifies the quasispherical components in the representations with minimal support over the rational cherednik algebras h m n s n and h n m s m in fact we show that the simple quotients of the corresponding quasispherical subalgebras are isomorphic as filtered algebras this symmetry was essentially established in 8 in the spherical case and in 24 in the case gcd m n 1 and it has a natural interpretation in terms of invariants of torus knots

central limit theorem for a stratonovich integral with malliavin calculus this is the publisher s version also available electronically from http projecteuclid org euclid aop 1372859768

consistency of likelihood estimation for gibbs point processes strong consistency of the maximum likelihood estimator mle for parametric gibbs point process models is established the setting is very general it includes pairwise pair potentials finite and infinite multibody interactions and geometrical interactions where the range can be finite or infinite the gibbs interaction may depend linearly or non linearly on the parameters a particular case being hardcore parameters and interaction range parameters as important examples we deduce the consistency of the mle for all parameters of the strauss model the hardcore strauss model the lennard jones model and the area interaction model

stable pairs and the homfly polynomial given a planar curve singularity we prove a conjecture of oblomkov shende relating the geometry of its hilbert scheme of points to the homfly polynomial of the associated algebraic link more generally we prove an extension of this conjecture due to diaconescu hua soibelman relating stable pair invariants on the conifold to the colored homfly polynomial of the algebraic link our proof uses wall crossing techniques to prove a blowup identity on the algebro geometric side we prove a matching identity for the colored homfly polynomials of a link using skein theoretic techniques

chern simons matrix models and stieltjes wigert polynomials employing the random matrix formulation of chern simons theory on seifert manifolds we show how the stieltjes wigert orthogonal polynomials are useful in exact computations in chern simons matrix models we construct a biorthogonal extension of the stieltjes wigert polynomials not available in the literature necessary to study chern simons matrix models when the geometry is a lens space we also study the relationship between stieltjes wigert and rogers szego polynomials and the corresponding equivalence with a unitary matrix model finally we give a detailed proof of a result that relates quantum dimensions with averages of schur polynomials in the stieltjes wigert ensemble

a contribution of the trivial connection to the jones polynomial and witten s invariant of 3d manifolds ii we extend the results of our previous paper 1 from knots to links by using a formula for the jones polynomial of a link derived recently by n reshetikhin we establish a relation between the parameters of this formula and the multivariable alexander polynomial this relation is illustrated by an example of a torus link we check that our expression for the alexander polynomial satisfies some of its basic properties finally we derive a link surgery formula for the loop corrections to the trivial connection contribution to witten s invariant of rational homology spheres

chern simons perturbation theory ii we study the perturbation theory for three dimensional chern simons quantum field theory on a general compact three manifold without boundary we show that after a simple change of variables the action obtained by brs gauge fixing in the lorentz gauge has a superspace formulation the basic properties of the propagator and the feynman rules are written in a precise manner in the language of differential forms using the explicit description of the propagator singularities we prove that the theory is finite finally the anomalous metric dependence of the 2 loop partition function on the riemannian metric which was introduced to define the gauge fixing can be cancelled by a local counterterm as in the 1 loop case in fact the counterterm is equal to the chern simons action of the metric connection normalized precisely as one would expect based on the framing dependence of witten s exact solution invited talk at the xxth conference on differential geometric methods in physics new york june 1991

phase transition in continuum potts models we establish phase transitions for a class of continuum multi type particle systems with finite range repulsive pair interaction between particles of different type this proves an old conjecture of lebowitz and lieb a phase transition still occurs when we allow a background pair interaction between all particles which is superstable and has sufficiently short range of repulsion our approach involves a random cluster representation analogous to the fortuin kasteleyn representation of the potts model in the course of our argument we establish the existence of a percolation transition for gibbsian particle systems with random edges between the particles and also give an alternative proof for the existence of gibbs measures with supperstable interaction

a note on the uniqueness result for the inverse henderson problem the inverse henderson problem of statistical mechanics is the theoretical foundation for many bottom up coarse graining techniques for the numerical simulation of complex soft matter physics this inverse problem concerns classical particles in continuous space which interact according to a pair potential depending on the distance of the particles roughly stated it asks for the interaction potential given the equilibrium pair correlation function of the system in 1974 henderson proved that this potential is uniquely determined in a canonical ensemble and he claimed the same result for the thermodynamical limit of the physical system here we provide a rigorous proof of a slightly more general version of the latter statement using georgii s variant of the gibbs variational principle the inverse henderson problem of statistical mechanics is the theoretical foundation for many bottom up coarse graining techniques for the numerical simulation of complex soft matter physics this inverse problem concerns classical particles in continuous space which interact according to a pair potential depending on the distance of the particles roughly stated it asks for the interaction potential given the equilibrium pair correlation function of the system in 1974 henderson proved that this potential is uniquely determined in a canonical ensemble and he claimed the same result for the thermodynamical limit of the physical system here we provide a rigorous proof of a slightly more general version of the latter statement using georgii s variant of the gibbs variational principle

the yang baxter equation and invariants of links on considere le fait que l on peut construire des invariants d isotopie p et f pour des liens en utilisant certaines solutions des equations de yang baxter

on the hecke algebras and the colored homfly polynomial the colored homfly polynomial is the quantum invariant of oriented links in s 3 associated with irreducible representations of the quantum group u q sl n in this paper using an approach to calculate quantum invariants of links via the cabling projection rule we derive a formula for the colored homfly polynomial in terms of the characters of the hecke algebras and schur polynomials the technique leads to a fairly simple formula for the colored homfly polynomial of torus links this formula allows us to test the labastida marino vafa conjecture which reveals a deep relationship between chern simons gauge theory and string theory on torus links

bps states torus links and wild character varieties a string theoretic framework is constructed relating the cohomology of wild character varieties to refined stable pair theory and torus link invariants explicit conjectural formulas are derived for wild character varieties with a unique irregular point on the projective line for this case this leads to a conjectural colored generalization of existing results of hausel mereb and wong as well as shende treumann and zaslow