invariants of 3 manifolds via link polynomials and quantum groups the aim of this paper is to construct new topological invariants of compact oriented 3 manifolds and of framed links in such manifolds our invariant of a link in a closed oriented 3 manifold is a sequence of complex numbers parametrized by complex roots of 1 for a framed link in s 3 the terms of the sequence are equale to the values of the suitably parametrized jones polynomial of the link in the corresponding roots of 1 in the case of manifolds with boundary our invariant is a sequence of finite dimensional complex linear operators this produces from each root of unity q a 3 dimensional topological quantum field theory

asymptotic expansions of witten reshetikhin turaev invariants for some simple 3 manifolds the link l this paper investigates the invariant z m 0 when g sk for a simple family of rational homology 3 spheres obtained by integer surgery around 2 n type torus knots in particular we find a closed formula for a formal power series z m eq h in h q 1 from which zf m 0 may be derived for all sufficiently large primes r we show that this formal power series may be viewed as the asymptotic expansion around 4 1 of a multivalued holomorphic function of q with 1 contained on the boundary of its domain of definition for these particular manifolds most of which are not z homology spheres this extends work of ohtsuki and murakami in which the existence of power series with rational coefficients related to z m 0 was demonstrated for rational homology spheres the coefficients in the formal power series z m are expected to be identical to those obtained from a perturbative expansion of the witten chem simons path integral formula for z m 0 0 199 5 american institute of physics

the f q mock theta function conjecture and partition ranks in 1944 freeman dyson initiated the study of ranks of integer partitions here we solve the classical problem of obtaining formulas for ne n resp no n the number of partitions of n with even resp odd rank thanks to rademacher s celebrated formula for the partition function this problem is equivalent to that of obtaining a formula for the coefficients of the mock theta function f q a problem with its own long history dating to ramanujan s last letter to hardy little was known about this problem until dragonette in 1952 obtained asymptotic results in 1966 g e andrews refined dragonette s results and conjectured an exact formula for the coefficients of f q by constructing a weak maass poincare series whose holomorphic part is q 1f q24 we prove the andrews dragonette conjecture and as a consequence obtain the desired formulas for ne n and no n

series representations for some mathematical constants abstract the authors apply a classical series identity involving the psi or digamma function with a view to deriving series representations for a number of known mathematical constants several closely related consequences and results are also considered

polynomial time counting and sampling of two rowed contingency tables in this paper a markov chain for contingency tables with two rows is defined the chain is shown to be rapidly mixing using the path coupling method we prove an upper bound for the mixing time of the chain the upper bound is quadratic in the number of columns and linear in the logarithm of the table sum by considering a specific problem we prove a lower bound for the mixing time of the chain the lower bound is quadratic in the number of columns and linear in the logarithm of the number of columns a fully polynomial randomised approximation scheme for this problem is described

a large k asymptotics of witten s invariant of seifert manifolds we calculate a largek asymptotic expansion of the exact surgery formula for witten ssu 2 invariant of some seifert manifolds the contributions of all flat connections are identified an agreement with the 1 loop formula is checked a contribution of the irreducible connections appears to contain only a finite number of terms in the asymptotic series a 2 loop correction to the contribution of the trivial connection is found to be proportional to casson s invariant

rapidly mixing markov chains for sampling contingency tables with a constant number of rows we consider the problem of sampling almost uniformly from the set of contingency tables with given row and column sums when the number of rows is a constant cryan and dyer j comput system sci 67 2003 pp 291 310 have recently given a fully polynomial randomized approximation scheme fpras for the related counting problem which employs markov chain methods indirectly they leave open the question as to whether a natural markov chain on such tables mixes rapidly here we show that the 2 x 2 heat bath markov chain is rapidly mixing we prove this by considering first a heat bath chain operating on a larger window using techniques developed by morris random walks in convex sets ph d thesis department of statistics university of california berkeley ca 2000 and morris and sinclair siam j comput 34 2004 pp 195 226 for the multidimensional knapsack problem we show that this chain mixes rapidly we then apply the comparison method of diaconis and saloff coste ann appl probab 3 1993 pp 696 730 to show that the 2 x 2 chain is also rapidly mixing

balanced3ϕ2summation theorems foru n basic hypergeometric series abstract in this paper we begin the theory and application of the u n 1 generalization of the classical bailey transform and bailey lemma we work in the setting of multiple basic hypergeometric series very well poised on unitary groups u n 1 the u n 1 bailey transform is obtained from a suitably modified u n 1 terminating very well poised 4 3 summation theorem and termwise transformations it is then interpreted as a matrix inversion result for two infinite lower triangular matrices this provides a higher dimensional generalization of andrews matrix inversion formulation of the bailey transform as in the classical case the concept of a u n 1 bailey pair is introduced and then inverted this u n 1 inversion enables us to derive u n 1 terminating balanced 3 2 summations directly from suitable u n 1 terminating very well poised 6 5 summations and vice versa these pairs of u n 1 dual identities extend the classical one variable case of andrews special limiting cases of the u n 1 terminating balanced 3 2 summation yield u n 1 q gauss summations several u n 1 q chu vandermonde summations u n 1 q binomial theorems and a u n 1 cauchy identity many other consequences of the u n 1 6 2 and 3 2 summations and the u n 1 bailey transform are further developed in our subsequent papers these include a derivation of the u n 1 bailey lemma several terminating u n 1 q whipple transformations u n 1 10 9 transformations and nonterminating u n 1 q whipple transformations the classical case of all this work corresponding to a 1 or equivalently u 2 contains a substantial amount of the theory and application of one variable basic hypergeometric series

approach to equilibrium of glauber dynamics in the one phase region i the attractive case various finite volume mixing conditions in classical statistical mechanics are reviewed and critically analyzed in particular somefinite size conditions are discussed together with their implications for the gibbs measures and for the approach to equilibrium of glauber dynamics inarbitrarily large volumes it is shown that dobrushin shlosman s theory ofcomplete analyticity and its dynamical counterpart due to stroock and zegarlinski cannot be applied in general to the whole one phase region since it requires mixing properties for regions ofarbitrary shape an alternative approach based on previous ideas of oliveri and picco is developed which allows to establish results on rapid approach to equilibrium deeply inside the one phase region in particular in the ferromagnetic case we considerably improve some previous results by holley and aizenman and holley our results are optimal in the sene that for example they show for the first time fast convergence of the dynamicsfor any temperature above the critical one for thed dimensional ising model with or without an external field in part ii we extensively consider the general case not necessarily attractive and we develop a new method based on renormalizations group ideas and on an assumption of strong mixing in a finite cube to prove hypercontractivity of the markov semigroup of the glauber dynamics

on q series identities for false theta series abstract we prove several infinite families of q series identities for false theta functions and related series these identities are motivated by considerations of characters of modules of vertex operator superalgebras and of quantum dilogarithms we also obtain closely related modular identities of the gollnitz gordon andrews type as a byproduct of our identities we establish several identities for the rogers dilogarithm function coming from multi q hypergeometric series with double poles

q series identities and values of certain l functions as usual define dedekind s eta function z by the infinite product eta z q 1 24 prod n 1 infty big 1 q n big quad big q e 2 pi i z text throughout big in a recent paper d zagier proved that note empty products equal 1 throughout sum n 0 infty big eta 24z q big 1 q 24 big big 1 q 48 big cdots big 1 q 24n big big eta 24z d q e q where the series d q and e q are defined by begin align d q frac 1 2 sum n 1 infty frac q 24n 1 q 24n frac 1 2 sum n 1 infty d n q 24n frac 1 2 q 24 2q 48 2q 72 3q 96 cdots quad e q frac 1 2 sum n 1 infty bigg frac 12 n bigg nq n 2 frac 1 2 q frac 5 2 q 25 frac 7 2 q 49 frac 11 2 q 121 cdots quad end align here d n denotes the number of positive divisors of n we obtain two infinite families of such identities and describe some consequences for l functions and partitions for example if 2 is the kronecker character for sqrt 2 these identities can be used to show that begin align 2e t 8 sum n 0 infty frac big 1 e 2t big big 1 e 4t big cdots big 1 e 2nt big big 1 e t big big 1 e 3t big cdots big 1 e 2n 1 t big hspace 110pt sum n 0 infty bigg frac 1 8 bigg n cdot l chi 2 2n 1 cdot frac t n n end align

residue formulas for the large k asymptotics of witten s invariants of seifert manifolds the case of su 2 we derive the largek asymptotics of the surgery formula forsu 2 witten s invariants of general seifert manifolds the contributions of connected components of the moduli space of flat connections are identified the contributions of irreducible connections are presented in the residue form this allows us to express them in terms of intersection numbers on their moduli spaces

quantum invariant for torus link and modular forms we consider an asymptotic expansion of kashaev s invariant or of the colored jones function for the torus link t 2 2m we shall give q series identity related to these invariants and show that the invariant is regarded as a limit of q being n th root of unity of the eichler integral of a modular form of weight 3 2 which is related to the character

representations of mock theta functions abstract motivated by the works of liu we provide a unified approach to find appell lerch series and hecke type series representations for mock theta functions we establish a number of parameterized identities with two parameters a and b specializing the choices of a b we not only give various known and new representations for the mock theta functions of orders 2 3 5 6 and 8 but also present many other interesting identities we find that some mock theta functions of different orders are related to each other in the sense that their representations can be deduced from the same a b parameterized identity furthermore we introduce the concept of false appell lerch series we then express the appell lerch series false appell lerch series and hecke type series in this paper using the building blocks m x q z and f a b c x y q introduced by hickerson and mortenson as well as m x q z and f a b c x y q introduced in this paper we also show the equivalences of our new representations for several mock theta functions and the known representations

bailey pairs and indefinite quadratic forms we construct classes of bailey pairs where the exponent of q in n is an indefinite quadratic form as an application we obtain families of q hypergeometric mock theta multisums

computer calculation of witten s 3 manifold invariant witten s 2 1 dimensional chern simons theory is exactly solvable we compute the partition function a topological invariant of 3 manifolds on generalized seifert spaces thus we test the path integral using the theory of 3 manifolds in particular we compare the exact solution with the asymptotic formula predicted by perturbation theory we conclude that this path integral works as advertised and gives an effective topological invariant

quantum invariants modular forms and lattice points ii we study the su 2 witten reshetikhin turaev wrt invariant for the seifert fibered homology spheres with m exceptional fibers we show that the wrt invariant can be written in terms of differential of the eichler integrals of modular forms with weight 1 2 and 3 2 by use of nearly modular property of the eichler integrals we shall obtain asymptotic expansions of the wrt invariant in the large n limit we further reveal that the number of the gauge equivalent classes of flat connections which dominate the asymptotics of the wrt invariant in n is related to the number of integral lattice points inside the m dimensional tetrahedron

ramanujan s lost notebook vi the mock theta conjectures 1 introduction in this paper we shall consider only ramanujan s two families of fifth order mock theta functions these functions were briefly described in ramanujan s last letter to g h hardy 11 pp 354 3551 and g n watson subsequently 12 131 proved all the assertions about these functions contained in the letter subsequently the identities watson proved were greatly generalized in 2 also the fifth order mock theta functions were shown to have double series expansions involving indefinite quadratic forms 7 however there remains a profound mystery about these functions namely no one including ramanujan has ever proved that these functions are indeed mock theta functions and not just some clever combination of theta functions this problem was described in detail by watson 12 p 2741 and redescribed in 7 pp 113 l 141 in s p 97 eqs 3 5 and 3 6 a formula from the lost notebook was pointed out that would if valid lead to the establishment of at least some of the fifth order mock theta functions as truly mock theta functions in the sense of watson 12 p 78 footnote there are indeed ten such identities live for each of the two families we have two objects in this paper first we wish to show that the identities in each family are equivalent i e if one is true all five are true and if one is false all five are false to introduce our second objective we recall some basic notions from partition theory 4 p 142 8 p 841 the rank of a partition is the largest

chern simons witten invariants of lens spaces and torus bundles and the semiclassical approximation we derive explicit formulas for the chern simons witten invariants of lens spaces and torus bundles overs1 for arbitrary values of the levelk most of our results are for the groupg su 2 though some are for more general compact groups we explicitly exhibit agreement of the limiting values of these formulas ask with the semiclassical approximation predicted by the chern simons path integral

certain classes of series involving the zeta function the authors apply the theory of the double gamma function which was recently revived in the study of determinants of laplacians to evaluate some families of series involving the riemann zeta function introducing a presumably new mathematical constant in the theory of the double gamma function they also systematically evaluate a definite integral of the double gamma function and various classes of series associated with zeta functions some of these definite integrals are expressed in terms of quotients of double gamma functions

simultaneous analysis of several models in the three dimensional ising universality class we investigate several three dimensional lattice models believed to be in the ising universality class by means of monte carlo methods and finite size scaling these models include spin 1 models with nearestneighbor interactions on the simple cubic and on the diamond lattice for the simple cubic lattice we also include models with third neighbor interactions of varying strength and some equivalent neighbor models also included are a spin 1 model and a hard core lattice gas separate analyses of the numerical data confirm the ising like critical behavior of these systems on this basis we analyze all these data simultaneously such that the universal parameters occur only once this leads to an improved accuracy the thermal magnetic and irrelevant exponents are determined as y t51 5868 3 y h52 4816 1 and y i520 821 5 respectively the binder ratio is estimated as q5 m 2 2 m 4 50 62 341 4

on the quantum invariants for the spherical seifert manifolds we study the witten reshetikhin turaev su 2 invariant for the seifert manifolds s3 gamma where is a finite subgroup of su 2 we show that the wrt invariants can be written in terms of the eichler integral of modular forms with half integral weight and we give an exact asymptotic expansion of the invariants by use of the nearly modular property of the eichler integral we further discuss that those modular forms have a direct connection with the polyhedral group by showing that the invariant polynomials of modular forms satisfy the polyhedral equations associated to

the critical probability for random voronoi percolation in the plane is 1 2 we study percolation in the following random environment let z be a poisson process of constant intensity on 2 and form the voronoi tessellation of 2 with respect to z colour each voronoi cell black with probability p independently of the other cells we show that the critical probability is 1 2 more precisely if p 1 2 then the union of the black cells contains an infinite component with probability 1 while if p 1 2 then the distribution of the size of the component of black cells containing a given point decays exponentially these results are analogous to kesten s results for bond percolation in 2 the result corresponding to harris theorem for bond percolation in 2 is known zvavitch noted that one of the many proofs of this result can easily be adapted to the random voronoi setting for kesten s results none of the existing proofs seems to adapt the methods used here also give a new and very simple proof of kesten s theorem for 2 we hope they will be applicable in other contexts as well

binary tree summation monte carlo simulation for potts models abstract in this talk we briefly comment on sweeny and gliozzi methods cluster monte carlo method and recent transition matrix monte carlo for potts models we mostly concentrate on a new algorithm known as binary tree summation some of the most interesting features of this method will be highlighted such as simulating fractional number of potts states as well as offering the partition function and thermodynamic quantities as functions of temperature in a single run

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

dyson s ranks and maass forms which were defined by ramanujan and watson decades ago in his last letter to hardy dated january 1920 see pages 127 131 of 27 ramanujan lists 17 such functions and he gives 2 more in his lost notebook 27 in his paper the final problem an account of the mock theta functions 32 watson defines 3 further functions surprisingly much remains unknown about these enigmatic series ramanujan s claims about their analytic properties remain open and there is even debate concerning the rigorous definition of such a function despite these seemingly problematic issues ramanujan s mock theta functions indeed possess many striking properties and they have been the subject of an astonishing number of important works for example see 5 6 7 8 12 13 14 18 19 20 23 27 28 32 33 35 36 to name a few watson predicted this high level of activity in his 1936 presidential address to the london mathematical society with his prophetic words see page 80 of 32

on the explicit construction of higher deformations of partition statistics the modularity of the partition generating function has many important consequences for example asymptotics and congruences for p n in a series of papers the author and ono 12 13 connected the rank a partition statistic introduced by dyson to weak maass forms a new class of functions which are related to modular forms and which were rst considered in 11 here we do a further step towards understanding how weak maass forms arise from interesting partition statistics by placing certain 2 marked durfee symbols introduced by andrews 1 into the framework of weak maass forms to do this we construct a new class of functions which we call quasiweak maass forms because they have quasimodular forms as components as an application we prove two conjectures of andrews it seems that this new class of functions will play an important role in better understanding weak maass forms of higher weight themselves and also their derivatives as a side product we introduce a new method which enables us to prove transformation laws for generating functions over incomplete lattices

a certain class of rapidly convergent series representations for z 2n 1 abstract for a natural number n the authors propose and develop three new series representations for the riemann zeta function 2n 1 the infinite series occurring in each of these three representations for 2n 1 converges remarkably faster than that in wilton s result furthermore one of the three series representations for 2n 1 involves the most rapidly convergent series among all the hitherto known members of the family of series representations considered here relevant connections of the results presented in this paper with many other known series representations for 2n 1 are also briefly indicated

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

the bal and bcl bailey transform and lemma we announce a higher dimensional generalization of the bailey transform bailey lemma and iterative bailey chain concept in the setting of basic hypergeometric series very well poised on unitary a or symplectic c groups the classical case corresponding to a 1 or equivalently u 2 contains an immense amount of the theory and application of one variable basic hypergeometric series including elegant proofs of the rogers ramanujan schur identities

three manifold invariants and their relation with the fundamental group we consider the 3 manifold invariant i m which is defined by means of the chern simons quantum field theory and which coincides with the reshetikhin turaev invariant we present some arguments and numerical results supporting the conjecture that for nonvanishing i m the absolute value i m only depends on the fundamental group 1 m of the manifold m for lens spaces the conjecture is proved when the gauge group is su 2 in the case in which the gauge group is su 3 we present numerical computations confirming the conjecture

on the coupling time of the heat bath process for the fortuin kasteleyn random cluster model we consider the coupling from the past implementation of the random cluster heat bath process and study its random running time or coupling time we focus on hypercubic lattices embedded on tori in dimensions one to three with cluster fugacity at least one we make a number of conjectures regarding the asymptotic behaviour of the coupling time motivated by rigorous results in one dimension and monte carlo simulations in dimensions two and three amongst our findings we observe that for generic parameter values the distribution of the appropriately standardized coupling time converges to a gumbel distribution and that the standard deviation of the coupling time is asymptotic to an explicit universal constant multiple of the relaxation time perhaps surprisingly we observe these results to hold both off criticality where the coupling time closely mimics the coupon collector s problem and also at the critical point provided the cluster fugacity is below the value at which the transition becomes discontinuous finally we consider analogous questions for the single spin ising heat bath process

glauber dynamics on the cycle is monotone we study heat bath glauber dynamics for the ferromagnetic ising model on a finite cycle a graph where every vertex has degree two we prove that the relaxation time 2 is an increasing function of any of the couplings j xy we also prove some further inequalities and obtain exact asymptotics for 2 at low temperatures

efficient simulation of the random cluster model the simulation of spin models close to critical points of continuous phase transitions is heavily impeded by the occurrence of critical slowing down a number of cluster algorithms usually based on the fortuin kasteleyn representation of the potts model and suitable generalizations for continuous spin models have been used to increase simulation efficiency the first algorithm making use of this representation suggested by sweeny in 1983 has not found widespread adoption due to problems in its implementation however it has been recently shown that it is indeed more efficient in reducing critical slowing down than the more well known algorithm due to swendsen and wang here we present an efficient implementation of sweeny s approach for the random cluster model using recent algorithmic advances in dynamic connectivity algorithms

decay of correlations in random cluster models we prove exponential decay for the tail of the radius r of the cluster at the origin for subcritical random cluster models under an assumption slightly weaker than that here d is the number of dimensions specifically if throughout the subcritical phase then for some 0 this implies the exponential decay of the two point correlation function of subcritical potts models subject to a hypothesis of at least polynomial decay of this function similar results are known already for percolation and ising models and for potts models when the number q of available states is sufficiently large indeed the hypothesis of polynomial decay has been proved rigorously for these cases in two dimensions the hypothesis that is weaker than requiring that the susceptibility be finite i e that the two point function be summable the principal new technique is a form of russo s formula for random cluster models reported by bezuidenhout grimmett and kesten for the current application this leads to an analysis of a first passage problem for random cluster models and a proof that the associated time constant is strictly positive if and only if the tail of r decays exponentially

instanton homology of seifert fibred homology three spheres for an oriented integral homology 3 sphere 2 a casson has introduced an integer invariant a 2 that is defined by using the space 2 of conjugacy classes of irreducible representations of 2 into su 2 see 1 this invariant a 2 can be computed from a surgery or heegard description of 2 and satisfies a 2 ju 2 mod 2 where i 2 is the kervaire milnor rochlin invariant of 2 this powerful new invariant was used to settle an outstanding problem in 3 manifold topology namely showing that if 2 is a homotopy 3 sphere then ju 2 o c taubes 17 utilizing gauge theoretic considerations has reinterpreted casson s invariant in terms of flat connections refining this approach a floer 13 has recently defined another invariant of 2 its instanton homology which takes the form of an abelian group 2 with a natural z8 grading that is an enhancement of a 2 in that

q identities for maass waveforms reinterpretation des theoremes de andrews dyson hickerson en termes de q identite pour une certaine forme d onde de maass

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

simulation of potts models with real q and no critical slowing down a monte carlo algorithm is proposed to simulate the ferromagnetic q state potts model for any real qg0 a single update is a random sequence of disordering and deterministic moves one for each link of the lattice a disordering move attributes a random value to the link regardless of the state of the system while in a deterministic move this value is a state function the relative frequency of these moves depends on the two parameters q and ensuremath beta 1 kt the algorithm is not affected by critical slowing down and the dynamical critical exponent z is exactly vanishing we simulate in this way a three dimensional potts model in the range 2lql3 for estimating the critical value q c where the thermal transition changes from second order to first order and find q c 2 620 ifmmode pm else textpm fi 0 005

critical ising on the square lattice mixes in polynomial time the ising model is widely regarded as the most studied model of spin systems in statistical physics the focus of this paper is its dynamic stochastic version the glauber dynamics introduced in 1963 and by now the most popular means of sampling the ising measure intensive study throughout the last three decades has yielded a rigorous understanding of the spectral gap of the dynamics on mathbb z 2 everywhere except at criticality while the critical behavior of the ising model has long been the focus for physicists mathematicians have only recently developed an understanding of its critical geometry with the advent of sle cle and new tools to study conformally invariant systems

series representations in the spirit of ramanujan abstract using an integral transform with a mild singularity we obtain series representations valid for specific regions in the complex plane involving trigonometric functions and the central binomial coefficient which are analogues of the types of series representations first studied by ramanujan over certain intervals on the real line we then study an exponential type series rapidly converging to the special values of l functions and the riemann zeta function in this way a new series converging to catalan s constant with geometric rate of convergence less than a quarter is deduced further evaluations of some series involving hyperbolic functions are also given

damage spreading and critical exponents for model a ising dynamics using damage spreading and heat bath dynamics we study the ising model in 2 and 3 dimensions with non conservative dynamics our algorithm differs in some important points from previous ones which makes it rather efficient we give estimates for the exponent z which seem to be the most precise published so far 2 172 0 006 for d 2 2 032 0 004 for d 3 we also give precise estimates of the exponent introduced by janssen et al z phys b 73 1989 539 and of analogous but in principle independent exponents we find surprisingly that some of the latter agree with and give an explanation for this

sweeny and gliozzi dynamics for simulations of potts models in the fortuin kasteleyn representation we compare the correlation times of the sweeny and gliozzi dynamics for two dimensional ising and three state potts models and the three dimensional ising model for the simulations in the percolation representation the results are also compared with swendsen wang and wolff cluster dynamics it is found that sweeny and gliozzi dynamics have essentially the same dynamical critical behavior contrary to gliozzi s claim phys rev e 66 016115 2002 the gliozzi dynamics has critical slowing down comparable to that of other cluster methods for the two dimensional ising model both sweeny and gliozzi dynamics give good fits to logarithmic size dependences for the correlation times for two dimensional three state potts model their dynamical critical exponent z is 0 49 0 01 the three dimensional ising model has z 0 37 0 02

on the two dimensional stochastic ising model in the phase coexistence region near the critical point we consider the two dimensional stochastic ising model in finite square with free boundary conditions at inverse temperature 0 and zero external field using duality and recent results of ioffe on the wulff construction close to the critical temperature we extend some of the results obtained by martinelli in the low temperature regime to any temperature below the critical one in particular we show that the gap in the spectrum of the generator of the dynamics goes to zero in the thermodynamic limit as an exponential of the side length of with a rate constant determined by the surface tension along one of the coordinate axes we also extend to the same range of temperatures the result due to shlosman on the equilibrium large deviations of the magnetization with free boundary conditions

dynamic critical behavior of the chayes machta algorithm for the random cluster model i two dimensions we study via monte carlo simulation the dynamic critical behavior of the chayes machta dynamics for the fortuin kasteleyn random cluster model which generalizes the swendsen wang dynamics for the q state potts ferromagnet to non integer q 1 we consider spatial dimension d 2 and 1 25 q 4 in steps of 0 25 on lattices up to 10242 and obtain estimates for the dynamic critical exponent z cm we present evidence that when 1 q 1 95 the ossola sokal conjecture z cm is violated though we also present plausible fits compatible with this conjecture we show that the li sokal bound z cm is close to being sharp over the entire range 1 q 4 but is probably non sharp by a power as a byproduct of our work we also obtain evidence concerning the corrections to scaling in static observables

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

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

the colored jones polynomials and the simplicial volume of a knot we show that the set of colored jones polynomials and the set of generalized alexander polynomials defined by akutsu deguchi and ohtsuki intersect non trivially moreover it is shown that the intersection is at least includes the set of kashaev s quantum dilogarithm invariants for links therefore kashaev s conjecture can be restated as follows the colored jones polynomials determine the hyperbolic volume for a hyperbolic knot modifying this we propose a stronger conjecture the colored jones polynomials determine the simplicial volume for any knot if our conjecture is true then we can prove that a knot is trivial if and only if all of its vassiliev invariants are trivial in 13 r m kashaev defined a family of complex valued link invariants indexed by integers n 2 using the quantum dilogarithm later he calculated the asymptotic behavior of his invariant and observed that for the three simplest hyperbolic knots it grows as exp vol k n 2 when n goes to the infinity where vol k is the hyperbolic volume of the complement of a knotk 14 this amazing result and his conjecture that the same also holds for any hyperbolic knot have been almost ignored by mathematicians since his definition of the invariant is too complicated though it uses only elementary tools the aim of this paper is to reveal his mysterious definition and to show that his invariant is nothing but a specialization of the colored jones polynomial the colored jones polynomial is defined for colored links each component is decorated with an irreducible representation of the lie algebra sl 2 c the original jones polynomial corresponds to the case that all the colors are identical to the 2 dimensional fundamental representation we show that kashaev s invariant with parameter n coincides with the colored jones polynomial in a certain normalization with every color the n dimensional representation evaluated at the primitive n th root of unity we have to normalize the colored jones polynomial so that the value for the trivial knot is one for otherwise it always vanishes on the other hand there are other colored polynomial invariants the generalized multivariable alexander polynomial defined by y akutsu t deguchi and t ohtsuki 1 they used the same lie algebra sl 2 c but a different hierarchy of representations their invariants are parameterized by c 1 parameters an integer date february 1 2008 1991 mathematics subject classification 57m25 57m50 17b37 81r50

multiplicative q hypergeometric series arising from real quadratic fields andrews dyson and hickerson showed that 2 q hypergeometric series going back to ramanujan are related to real quadratic fields which explains interesting properties of their fourier coefficients there is also an interesting relation of such series to automorphic forms here we construct more such examples arising from interesting combinatorial statistics

superconformal algebras and mock theta functions it is known that characters of bps representations of extended superconformal algebras do not have good modular properties due to extra singular vectors coming from the bps condition in order to improve their modular properties we apply the method of zwegers which has recently been developed to analyze modular properties of mock theta functions we consider the case of the superconformal algebra at general levels and obtain the decomposition of characters of bps representations into a sum of simple jacobi forms and an infinite series of non bps representations we apply our method to study elliptic genera of hyper kahler manifolds in higher dimensions in particular we determine the elliptic genera in the case of complex four dimensions of the hilbert scheme of points on k3 surfaces k 2 and complex tori a 3

applying coordinate products to the topological identification of normed spaces using the l 2 products we find pre hilbert spaces that are absorbing sets for all borelian classes of order 1 we also show that the following spaces are homeomorphic to the countable product of the space x n r x n is bounded 1 every coordinate product c h n of normed spaces h n in the sense of a banach space c where each h n is an absolute f set and infinitely many of the h n s are z spaces 2 every function space l p p p l p with the l q topology 0 q p 3 every sequence space l p p p l p with the l q topology 0 p q we also note that each additive and multiplicative borelian class of order 2 each projective class and the class of nonprojective spaces contain uncountably many topologically different pre hilbert spaces which are z spaces

graphical representations and cluster algorithms ii we continue the study initiated in part i of graphical representations and cluster algorithms for various models in or related to statistical mechanics for certain models e g the blume emery griffths model and various generalizations we develop fortuin kasteleyn type representations which lead immediately to swendsen wang type algorithms for other models e g the random cluster model that are defined by a graphical representation we develop cluster algorithms without reference to an underlying spin system in all cases phase transitions are related to percolation or incipient percolation in the graphical representation which via the ic algorithm allows for the rapid simulation of these systems at the transition point pertinent examples include the continuum widom rowlinson model the restricted 1 step solid on solid model and the xy model

overpartitions and class numbers of binary quadratic forms we show that the zagier eisenstein series shares its nonholomorphic part with certain weak maass forms whose holomorphic parts are generating functions for overpartition rank differences this has a number of consequences including exact formulas asymptotics and congruences for the rank differences as well as q series identities of the mock theta type

the swendsen wang process does not always mix rapidly the swendsen wang process provides one possible dynamics for the q state potts model computer simulations of this process are widely used to estimate the expectations of various observables random variables of a potts system in the equilibrium or gibbs distribution the legitimacy of such simulations depends on the rate of convergence of the process to equilibrium as measured by the mixing time empirical observations suggest that the mixing time of the swendsen wang process is short in many instances of practical interest although proofs of this desirable behavior are known only for some very special cases nevertheless we show that there are occasions when the mixing time of the swendsen wang process is exponential in the size of the system this undesirable behavior is related to the phenomenon of first order phase transitions in potts systems with q 2 states

structure and eigenvalues of heat bath markov chains we prove that heat bath chains which we define in a general setting have no negative eigenvalues two applications of this result are presented one to single site heat bath chains for spin systems and one to a heat bath markov chain for sampling contingency tables some implications of our main result for the analysis of the mixing time of heat bath markov chains are discussed we also prove an alternative characterisation of heat bath chains and consider possible generalisations

almost partition identities an almost partition identity is an identity for partition numbers that is true asymptotically formula see text of the time and fails infinitely often we prove a kind of almost partition identity namely that the number of parts in all self conjugate partitions of n is almost always equal to the number of partitions of n in which no odd part is repeated and there is exactly one even part possibly repeated not only does the identity fail infinitely often but also the error grows without bound in addition we prove several identities involving the number of parts in restricted partitions we show that the difference in the number of parts in all self conjugate partitions of n and the number of parts in all partitions of n into distinct odd parts equals the number of partitions of n in which no odd part is repeated the smallest part is odd and there is exactly one even part possibly repeated we provide both analytic and combinatorial proofs of this identity

ramanujan s lost notebook v euler s partition identity on examine deux identites de ramanujan etroitement liees a l identite d euler relative aux partitions