an observation on the partition function zeros of the hard hexagon model it is conjectured that part of the limiting locus of partition function zeros for the hard hexagon model is an algebraic curve generated by a simple and rational algebraic equation this equation arises from the same mathematical mechanism as that which determines algebraic invariant properties of self dual systems a remarkable approximation to the critical point of the three state potts model triangular lattice is mentioned in passing

conformal iterated function systems with applications to the geometry of continued fractions in this paper we obtain some results about general conformal iterated function systems we obtain a simple characterization of the packing dimension of the limit set of such systems and introduce some special systems which exhibit some interesting behavior we then apply these results to the set of values of real continued fractions with restricted entries we pay special attention to the hausdorff and packing measures of these sets we also give direct interpretations of these measure theoretic results in terms of the arithmetic density properties of the set of allowed entries

modular properties of the hard hexagon model the physical quantities or powers thereof in the hard hexagon model that were computed exactly by baxter are shown to be modular functions with respect to the number theoretic group 1 n this allows us to determine the analytic structure of the partition function per site in the thermodynamic limit and the density as functions of the activityz

duality between the trigonometric bcn sutherland system and a completed rational ruijsenaars schneider van diejen system we present a new case of duality between integrable many body systems where two systems live on the action angle phase spaces of each other in such a way that the action variables of each system serve as the particle positions of the other one our investigation utilizes an idea that was exploited previously to provide group theoretic interpretation for several dualities discovered originally by ruijsenaars in the group theoretic framework one applies hamiltonian reduction to two abelian poisson algebras of invariants on a higher dimensional phase space and identifies their reductions as action and position variables of two integrable systems living on two different models of the single reduced phase space taking the cotangent bundle of u 2n as the upstairs space we demonstrate how this mechanism leads to a new dual pair involving the bcn trigonometric sutherland system thereby we generalize earlier results pertaining to the an trigonometric sutherland system as well as a recent work by pusztai on the hyperbolic bcn sutherland system

lax representation of the hyperbolic van diejen dynamics with two coupling parameters in this paper we construct a lax pair for the classical hyperbolic van diejen system with two independent coupling parameters built upon this construction we show that the dynamics can be solved by a projection method which in turn allows us to initiate the study of the scattering properties as a consequence we prove the equivalence between the first integrals provided by the eigenvalues of the lax matrix and the family of van diejen s commuting hamiltonians also at the end of the paper we propose a candidate for the lax matrix of the hyperbolic van diejen system with three independent coupling constants

the exact location of partition function zeros a new method for statistical mechanics a new mathematical mechanism is proposed for locating exactly sections of the locus of the limiting distribution of partition function zeros for model systems in statistical mechanics illustrations and applications to the two dimensional ising model and the hard square lattice gas are given the exact location of the branch point singularity for the hard square lattice gas on the negative activity z axis is conjectured to be the negative root of the polynomial 2z6 9z5 42z4 90z3 96z2 27z 2

equation of state and isothermal compressibility for the hard hexagon model in the disordered regime the equation of state for the hard hexagon model in the disordered regime is computed the isothermal compressibility is given as a rational function of k and p the hard hexagon model is a lattice gas on a triangular lattice with the rule that if a site is occupied then the six neighbouring sites are necessarily vacant baxter 1980 1981 has shown that this model is exactly solvable by the method of commuting transfer matrices baxter 1972 1982 in the disordered regime baxter s exact results are

integrable spin chain with reflecting end a new integrable spin chain of the haldane shastry type is introduced it is interpreted as the inverse square interacting spin chain with a reflecting end the lattice sites of this model consist of the square roots of the zeros of the laguerre polynomial using the exchange operator formalism the integrals of motion for the model are explicitly constructed

quadratic algebra associated with rational calogero moser models classical calogero moser models with rational potential are known to be superintegrable that is on top of the r involutive conserved quantities necessary for the integrability of a system with r degrees of freedom they possess an additional set of r 1 algebraically and functionally independent globally defined conserved quantities at the quantum level kuznetsov uncovered the existence of a quadratic algebrastructure as an underlying key for superintegrability for the models based on a type root systems here we demonstrate in a universal way the quadratic algebrastructure for quantum rational calogero moser models based on any root systems

differential equations in the spectral parameter we determine all the potentialsv x for the schrodinger equation 2 v x k2 such that some family of eigenfunctions satisfies a differential equation in the spectral parameterk of the formb k k o x o for each suchv x we determine the algebra of all possible operatorsb and the corresponding functions x

on a point of order a method of using algebraic curves to obtain estimates of critical points accurate enough to identify them as simple algebraic numbers if that is what they are is discussed and illustrated with an application to the q state potts model on the triangular lattice for cases of pure two site interactions and pure three site interactions in the latter case the critical point is conjectured to be z c 2 tfrac 1 2 sqrt 3 q 2 2 q q geqslant in a similar conjecture for the critical percolation probability on thedirected square lattice q 1 2 q c 3 2 q c p c 1

the algebraic construction of partition function zeros universality and algebraic cycles the author considers the one parameter lattice models of classical statistical mechanics from a simple algebraic viewpoint the way in which the limiting locus of partition function zeros emerges through a sequence of semi infinite m infinity lattice sections is considered convergence to the limiting locus cinfinity is obtained through two sequences of algebraic curves for all arbitrarily large but finite m the partition function per site is a branch of an algebraic function lambda 1 defined by an irreducible polynomial and possesses only a finite number of algebraic singular points in the limit of m to infinity infinitely many branch points accumulate on cinfinity an algebraic basis to the universality hypothesis is presented in terms of the accumulation of branch points at the critical point it is argued that for many two dimensional lattice models the critical exponents alpha and nu will be of the form alpha 2 s and nu s 1 s where s is an integer or 3

fractal measures for parabolic ifs abstract let h be the hausdorff dimension of the limit set of a conformal parabolic iterated function system in dimension d 2 in case the system of maps is finite we provide necessary and sufficient conditions for the h dimensional hausdorff measure to be positive and finite and also assuming the strong open set condition holds characterize when the h dimensional packing measure of the limit set is positive and finite we also prove that the upper ball box counting dimension and the hausdorff dimension of this limit set coincide as a byproduct we include a compact analysis of the behaviour of parabolic conformal diffeomorphisms in dimension 2 and separately in any dimension greater than or equal to 3

numerical experiments on yang lee zeros it is argued that since any real space renormalisation r is a transformation under which the partition function is invariant those r which are based upon various real space approximation schemes can in principle be extended into the complex field of the parameter space in this extension the julia set of r now extended to include r which are algebraic is an approximation to the locus of the yang lee zeros finite size scaling using onsager s solution of the two dimensional ising model is used to construct r which are highly accurate in real temperatures in the critical region these are extended into the complex temperature plane and results are displayed for both the anisotropic and isotropic cases

hausdorff dimension of harmonic measure for self conformal sets abstract under some technical assumptions it is shown that the hausdorff dimension of the harmonic measure on the limit set of a conformal infinite iterated function system is strictly less than the hausdorff dimension of the limit set itself if the limit set is contained in a real analytic curve if the iterated function system consists of similarities only or if this system is irregular as a consequence of this general result the same statement is proven for hyperbolic and parabolic julia sets finite parabolic iterated function systems and generalized polynomial like mappings also sufficient conditions are provided for a limit set to be uniformly perfect and for the harmonic measure to have the hausdorff dimension less than 1 some results in the spirit of przytycki et al ann of math 130 1989 1 40 stud math 97 1991 189 225 are obtained

exact solution of a directed site animals enumeration problem in three dimensions an exact equivalence is established between the directed site animals enumeration dsae problem in d dimensions and a crystal growth model defined on the same lattice in special cases the latter reduces to the calculation of the free energy of a d ensuremath 1 dimensional lattice gas with extended hard cores the author solves a dsae problem with d 3 exactly by using its equivalence to the hard hexagon problem and shows that the exponent ensuremath theta frac 5 6 a new and simpler solution to the dsae problem on the square and triangular lattices is given

quantum versus classical integrability in calogero moser systems calogero moser systems are classical and quantum integrable multiparticle dynamics defined for any root system the quantum calogero systems having 1 q2 potential and a confining q2 potential and the sutherland systems with 1 sin2q potentials have integer energy spectra characterized by the root system various quantities of the corresponding classical systems e g minimum energy frequencies of small oscillations the eigenvalues of the classical lax pair matrices etc at the equilibrium point of the potential are investigated analytically as well as numerically for all root systems to our surprise most of these classical data are also integers or they appear to be quantized to be more precise these quantities are polynomials of the coupling constant s with integer coefficients the close relationship between quantum and classical integrability in calogero moser systems deserves fuller analytical treatment which would lead to better understanding of these systems and of integrable systems in general

location of zeros in the complex temperature plane absence of lee yang theorem in the yang lee theory 1952 the occurrence of a phase transition at real physical values of the magnetic field is related to the behaviour of the density of zeros of the partition function in the complex magnetic field plane as shown by lee and yang these zeros fall on lines for a wide class of ferromagnetic models in extensions of the yang lee theory to the complex temperature plane it has often been assumed that the zeros of the partition function would again fall on lines rather than in areas though this happens to be true for the isotropic ising model the authors point out that there is in general no basis for this assumption they discuss the location of the zeros of the partition function of the anisotropic ising model in the complex temperature plane and show that they indeed always fall in areas

series expansion of the percolation probability for the directed square lattice by extrapolation from finite lattices the authors extend the known series for the percolation probability on the directed square lattice from eight terms to 41 analysing the series they obtain the estimates qc 0 355 299 or 0 000 001 beta 0 2764 or 0 0001 for the critical probability and the critical exponent from this together with scaling relations and previous results on the moments of the pair connectedness function the authors conjecture that beta may be exactly 199 720

perimeter bethe ansatz the partition function of a finite z invariant six vertex lattice model with a prescribed arrow configuration at the boundary is given the expression is of the same type that occurs in the bethe ansatz but there are no wavenumber equations to solve

triangular lattice potts models in this paper we study the 3 state potts model on the triangular lattice which has two and three site interactions using a peierls argument we obtain a rigorous bound on the transition temperature thereby disproving a conjecture on the location of its critical point low temperature series are generated and analyzed for three particular choices of the coupling constants a phase diagram is then drawn on the basis of these considerations our analysis indicates that the antiferromagnetic transition and the transition along the coexistence line are of first order implying the existence of a multicritical point in the ferromagnetic region relation of the triangularq state potts model with other lattice statistical problems is also discussed in particular an ashkin teller model and the hard hexagon lattice gas solved by baxter emerge as special cases in appropriate limits

matrices differential operators and polynomials it is noted that the matrix z of order n defined in terms of the n arbitrary numbers xj by the formula zjk jk j l 1 xj xl 1 1 jk xj xk 1 may be considered in an appropriate framework to correspond to the differential operator d dx there follow prescriptions to construct explicit matrices of arbitrary order n in terms of n or more arbitrary numbers or of the n roots of given polynomials matrices whose eigenvalues and eigenvectors are given fully or in part by very simple formulas novel representations of the classical polynomials hermite laguerre lagrange gegenbauer jacobi are also obtained such as the formula for hermite polynomials hn x 2n det xi h where i is the unit matrix of order n and the matrix h of order n is defined by hjk 2n 1 2 jk n 1 exp 2i j 1 2 exp 2i j 1 jk exp 2i j 2i sin j k 1 exp i j k with j j n arbitrary

fractal structure of zeros in hierarchical models we consider an ising and aq state potts model on a diamond hierarchical lattice we give pictures of the distribution of zeros of the partition function in the complex plane of temperatures for several choices ofq these zeros are just the julia set corresponding to the renormalization group transformation

comment on the exact location of partition function zeros a new method for statistical mechanics in a recent paper wood 1985 conjectured that the exact location of the branch point singularity for the hard square lattice gas on the negative activity z axis is at z 0 119392 a careful analysis of the available series data implies that this conjecture is almost certainly false the author finds the correct value to be z 0 1193388809 or 0 000000001 with the corresponding exponent indistinguishably close to 5 6 corresponding to a zero of the singular part of the function

analytic properties of the partition function for statistical mechanical models the author explains with examples the relationship between zeros of the partition function analyticity and degeneracy of absolute magnitude of eigenvalues of the transfer matrix for statistical mechanical models it is shown how to write down a polynomial representation of the partition function for any model with a two sheet largest eigenvalue the staggered ice model representation of the q state potts model on a sequence of semi infinite strips is solved and it is shown that for 0 q 4 the potts model result is only projected out by special boundary conditions the non potts part of the result is obtained for a 4 infinity strip and gives baxter s antiferromagnetic critical curve 1982

action angle maps and scattering theory for some finite dimensional integrable systems i the pure soliton case we construct an action angle transformation for the calogero moser systems with repulsive potentials and for relativistic generalizations thereof this map is shown to be closely related to the wave transformations for a large classl of hamiltonians and is shown to have remarkable duality properties all dynamics inl lead to the same scattering transformation which is obtained explicitly and exhibits a soliton structure an auxiliary result concerns the spectral asymptotics of matrices of the formm exp td ast it pertains to diagonal matricesd whose diagonal elements have pairwise different real parts and to matricesm for which certain principal minors are non zero

exact potts model partition functions on ladder graphs abstract we present exact calculations of the partition function z of the q state potts model and its generalization to real q for arbitrary temperature on n vertex ladder graphs i e strips of the square lattice with width ly 2 and arbitrary length lx with free cyclic and mobius longitudinal boundary conditions these partition functions are equivalent to tutte whitney polynomials for these graphs the free energy is calculated exactly for the infinite length limit of these ladder graphs and the thermodynamics is discussed by comparison with strip graphs of other widths we analyze how the singularities at the zero temperature critical point of the ferromagnet on infinite length finite width strips depend on the width we point out and study the following noncommutativity at certain special values q s lim n lim q q s z 1 n lim q q s lim n z 1 n it is shown that the potts antiferromagnet on both the infinite length line and ladder graphs with cyclic or mobius boundary conditions exhibits a phase transition at finite temperature if 0 b in the corresponding c 2 space arising as the accumulation set of partition function zeros as n in particular we study the connection with the t 0 limit of the potts antiferromagnet where b reduces to the accumulation set of chromatic zeros certain properties of the complex temperature phase diagrams are shown to exhibit close connections with those of the model on the square lattice showing that exact solutions on infinite length strips provide a way of gaining insight into these complex temperature phase diagrams

rodrigues formulas for the macdonald polynomials we present formulas of rodrigues type giving the macdonald polynomials for arbitrary partitions through the repeated application of creation operatorsbk k 1 l on the constant 1 three expressions for the creation operators are derived one from the other when the last of these expressions is used the associated rodrigues formula readily implies the integrality of the q t kostka coefficients the proofs given in this paper rely on the connection between affine hecke algebras and macdonald polynomials

wave functions for quantum integrable particle systems via partial confluences of multivariate hypergeometric functions abstract starting from the hyperoctahedral multivariate hypergeometric function of heckman and opdam associated with the b c n root system we arrive via partial confluent limits in the sense of oshima and shimeno at solutions of the eigenvalue equations for the quantum toda chain with one sided boundary perturbations of poschl teller type and for the hyperbolic quantum calogero sutherland system in a morse potential with the aid of corresponding degenerations of the bispectral dual difference equations for the heckman opdam hyperoctahedral hypergeometric function it is deduced that the eigensolutions in question are holomorphic in the spectral variable

on the duality between the hyperbolic sutherland and the rational ruijsenaars schneider models we consider two families of commuting hamiltonians on the cotangent bundle of the group and show that upon an appropriate single symplectic reduction they descend to the spectral invariants of the hyperbolic sutherland and of the rational ruijsenaars schneider lax matrices respectively the duality symplectomorphism between these two integrable models that was constructed by ruijsenaars using direct methods can then be interpreted geometrically simply as a gauge transformation connecting two cross sections of the orbits of the reduction group

the calogero sutherland model and generalized classical polynomials multivariable generalizations of the classical hermite laguerre and jacobi polynomials occur as the polynomial part of the eigenfunctions of certain schrodinger operators for calogero sutherland type quantum systems for the generalized hermite and laguerre polynomials the multidimensional analogues of many classical results regarding generating functions differentiation and integration formulas recurrence relations and summation theorems are obtained we use this and related theory to evaluate the global limit of the ground state density obtaining in the hermite case the wigner semi circle law and to give an explicit solution for an initial value problem in the hermite and laguerre case

on the icosahedral equation and the locus of zeros for the grand partition function of the hard hexagon model the kleinian theory 1922 of the icosahedral equation is used to investigate a recent conjecture on the locus of zeros for the grand partition function of the hard hexagon lattice gas model

quantum calogero moser models integrability for all root systems the issues related to the integrability of quantum calogero moser models based on any root systems are addressed for the models with degenerate potentials i e the rational with without the harmonic confining force the hyperbolic and the trigonometric we demonstrate the following for all the root systems i construction of a complete set of quantum conserved quantities in terms of a total sum of the lax matrix l i e sigma mu nu is an element ofr l n mu nu in which r is a set of r r vectors invariant under the action of the coxeter group they form a single coxeter orbit ii proof of liouville integrability iii triangularity of the quantum hamiltonian and the entire discrete spectrum generalized jack polynomials are defined for all root systems as unique eigenfunctions of the hamiltonian iv equivalence of the lax operator and the dunkl operator v algebraic construction of all excited states in terms of creation operators these are mainly generalizations of the results known for the models based on the a series i e su n type root systems

integrability versus non integrability hard hexagons and hard squares compared in this paper we compare the integrable hard hexagon model with the nonintegrable hard squares model by means of partition function roots and transfer matrix eigenvalues we consider partition functions for toroidal cylindrical and free free boundary conditions up to sizes 40 40 and transfer matrices up to 30 sites for all boundary conditions the hard squares roots are seen to lie in a bounded area of the complex fugacity plane along with the universal hard core line segment on the negative real fugacity axis the density of roots on this line segment matches the derivative of the phase difference between the eigenvalues of largest and equal moduli and exhibits much greater structure than the corresponding density of hard hexagons we also study the special point z 1 of hard squares where all eigenvalues have unit modulus and we give several conjectures for the value at z 1 of the partition functions

algebraic approximations to the locus of partition function zeros applications of the mathematical formalism put forward in the previous paper see ibid p 3471 1987 are made to a number of well known lattice models in statistical mechanics sequences of algebraic curves are obtained from the branches of an algebraic function lambda 1 where lambda 1 at real temperatures is the branch of lambda 1 which is the partition function per site of an m infinity lattice section these sequences are viewed as approximations to parts of the limiting locus cinfinity of partition function zeros approximations to critical points can be obtained in a natural extension of these curves beyond the branch points of lambda 1

poisson lie interpretation of trigonometric ruijsenaars duality a geometric interpretation of the duality between two real forms of the complex trigonometric ruijsenaars schneider system is presented the phase spaces of the systems in duality are viewed as two different models of the same reduced phase space arising from a suitable symplectic reduction of the standard heisenberg double of u n the collections of commuting hamiltonians of the systems in duality are shown to descend from two families of free hamiltonians on the double which are dual to each other in a poisson lie sense our results give rise to a major simplification of ruijsenaars proof of the crucial symplectomorphism property of the duality map

zeros of the partition function for the two dimensional ising model the zeros of the partition function for the zero field two dimensional ising model in the complex temperature plane are represented in terms of the eigenvalues of the transfer matrix the boundaries of the limiting distribution have a particular simple representation in terms of the correlation range along the two principal axes of the quadratic lattice examples of the distribution in the complex plane of z exp 2k are given for the anisotropic model when the horizontal and vertical interactions are in integer ratio

conformal geometric and invariant measures for transcendental expanding functions we describe the fractal structure of expanding meromorphic maps of the form h circ exp circ q where h and q are rational functions whose most transparent examples are among the functions of the form frac a exp z p b exp z p c exp z p d exp z p with ad bc ne 0 in particular we show that depending upon whether the hausdorff dimension of the julia set is greater or less than 1 the finite non zero geometric measure is provided by the hausdorff or packing measure in order to describe this fractal structure we introduce and explore in detail walters expanding conformal maps and jump like conformal maps

universal lax pairs for spin calogero moser models and spin exchange models for any root system and a set of vectors which form a single orbit of the reflection weyl group g generated by a spin calogero moser model can be defined for each of the potentials rational hyperbolic trigonometric and elliptic for each member of to be called a site we associate a vector space v whose element is called a spin its dynamical variables are the canonical coordinates of a particle in rr r rank of and spin exchange operators which exchange the spins at the sites and s here s is the reflection generated by for each and a spin exchange model can be defined the hamiltonian of a spin exchange model is a linear combination of the spin exchange operators only it is obtained by freezing the canonical variables at the equilibrium point of the corresponding classical calogero moser model for ar and set of vector weights it reduces to the well known haldane shastry model universal lax pair operators for both spin calogero moser models and spin exchange models are presented which enable us to construct as many conserved quantities as the number of sites for degenerate potentials

double coset decompositions of reductive lie groups arising from two involutions g 0 h g and g 0 l g here g g g g g for an automorphism of g and f0 denote the connected component of f containing the identity e for a lie group f in 8 we gave standard representatives of double coset decompositions h g l for some typical triples g h l by only using algebraic method as the most elementary example we studied the case g h l gl n if gl p if gl n p if gl r if gl n r if for an arbitrary field if we also studied in 8 other examples related to quadratic forms on vector spaces over the fields ir c and ih in this paper we assume that g is a reductive lie group and we will describe the structure of the double coset decomposition h g l for an arbitrary g h l first we study the compact case section 3 and next the noncompact case section 4 in section 3 we assume that the semisimple part gs of g is compact gs is the analytic subgroup of g for gs g g when g gs and

chromatic polynomials of large triangular lattices evaluating the q colourings of a lattice is equivalent to solving the q state zero temperature antiferromagnetic potts model this has recently been done exactly for an infinite triangular lattice with q real here the results are extended to the full complex q plane giving the limiting distribution of the zeros of the chromatic polynomial the results are compared with finite lattice calculations and the occurrence of isolated real zeros converging on the beraha numbers is noted

differential difference operators associated to reflection groups there is a theory of spherical harmonics for measures invariant under a finite reflection group the measures are products of powers of linear functions whose zero sets are the mirrors of the reflections in the group times the rotation invariant measure on the unit sphere in rn a commutative set of differential difference operators each homogeneous of degree 1 is the analogue of the set of first order partial derivatives in the ordinary theory of spherical harmonics in the case of r2 and dihedral groups there are analogues of the cauchy riemann equations which apply to gegenbauer and jacobi polynomial expansions the analysis of orthogonality structures for polynomials in several variables is a problem of vast dimensions this paper is part of an ongoing program to establish a workable theory for one particular class the underlying structure is based on finite coxeter groups these are finite groups acting on euclidean space generated by reflections in the zero sets of a collection of linear functions the roots the weight functions for the orthogonality are products of powers of these linear functions restricted to the surface of the unit sphere in addition the weight function is required to be invariant under the action of the group the resulting theory has strong similarities to the theory of spherical harmonics this was established in previous papers of the author 3 4 5 most notably a homogeneous polynomial is orthogonal to all polynomials of lower degree if and only if it is annihilated by a certain second order differential difference operator ordinary partial differentiation acts as an endomorphism on ordinary harmonic functions the use of such operators and their adjoints leads to recurrence formulas and orthogonal decompositions for harmonic polynomials in this paper we construct a commutative set of first order differentialdifference operators associated to the second order operator previously mentioned received by the editors june 1 1987 1980 mathematics subject classification 1985 revision primary 33a45 33a65 20h15 secondary 20c30 42c10 51f15

difference equation for the heckman opdam hypergeometric function and its confluent whittaker limit abstract we present an explicit difference equation for the heckman opdam hypergeometric function associated with root systems via a confluent hypergeometric limit an analogous difference equation is obtained for the class one whittaker function diagonalizing the open quantum toda chain

geometric rigidity of transcendental meromorphic functions the classes of dynamically and geometrically tame functions meromorphic outside a small set are introduced the julia sets of geometrically tame functions are proven to be either geometrical circle in open image in new window or to have hausdorff dimension strictly larger than 1 vast classes of dynamically and geometrically tame functions are identified

trigonometric sutherland systems and their ruijsenaars duals from symplectic reduction beside its usual interpretation as a system of n indistinguishable particles moving on the circle the trigonometric sutherland system can be viewed alternatively as a system of distinguishable particles on the circle or on the line and these three physically distinct systems are in duality with corresponding variants of the rational ruijsenaars schneider system we explain that the three duality relations first obtained by ruijsenaars in 1995 arise naturally from the kazhdan kostant sternberg symplectic reductions of the cotangent bundles of the group u n and its covering groups u 1 su n and r su n respectively this geometric interpretation enhances our understanding of the duality relations and simplifies ruijsenaars original direct arguments that led to their discovery

a new model in the calogero ruijsenaars family hamiltonian reduction is used to project a trivially integrable system on the heisenberg double of su n n to obtain a system of ruijsenaars type on a suitable quotient space this system possesses bcn symmetry and is shown to be equivalent to the standard three parameter bcn hyperbolic sutherland model in the cotangent bundle limit

self duality and scattering map for the hyperbolic van diejen systems with two coupling parameters with an appendix by s ruijsenaars in this paper we construct global action angle variables for a certain two parameter family of hyperbolic van diejen systems following ruijsenaars ideas on the translation invariant models the proposed action angle variables come from a thorough analysis of the commutation relation obeyed by the lax matrix whereas the proof of their canonicity is based on the study of the scattering theory as a consequence we show that the van diejen system of our interest is self dual with a factorized scattering map also in an appendix by s ruijsenaars a novel proof of the spectral asymptotics of certain exponential type matrix flows is presented this result is of crucial importance in our scattering theoretical analysis

common algebraic structure for the calogero sutherland models we investigate a common algebraic structure for the rational and trigonometric calogero sutherland models by using the exchange operator formalism we show that the set of jack polynomials whose arguments are dunkl type operators provides an orthogonal basis for the rational case

critical point of a triangular potts model with two and three site interactions the q state potts model on the triangular lattice with nearest neighbour interactions and three site interactions in half of the triangular faces is considered the exact duality relation is re examined from the point of view of determining its critical point using the continuity and uniqueness arguments the authors determine the exact critical point in the ferromagnetic model it is argued that a transition exists in an antiferromagnetic model only for q 3 a conjecture is then made on the phase diagram for the q 3 isotropic model these results are used to determine the exact criticality of a dilute potts model on the honeycomb lattice

z2 11z 1 as an algebraic invariant for the hard hexagon model the limiting partition function per site xi infinity z for the hard hexagon model has two singular points at the roots of z2 11z 1 0 these two points appear to be algebraic invariants associated with algebraic functions defined on an infinite sequence of finite transfer matrices

dimensions and measures in infinite iterated function systems the hausdorff and packing measures and dimensions of the limit sets of iterated function systems generated by countable families of conformal contractions are investigated conformal measures for such systems reflecting geometric properties of the limit set are introduced proved to exist and to be unique the existence of a unique invariant probability equivalent to the conformal measure is derived our methods employ the concepts of the perron frobenius operator symbolic dynamics on a shift space with an infinite alphabet and the properties of the above mentioned ergodic invariant measure a formula for the hausdorff dimension of the limit set in terms of the pressure function is derived fractal phenomena not exhibited by finite systems are shown to appear in the infinite case in particular a variety of conditions are provided for hausdorff and packing measures to be positive or finite and a number of examples are described showing the appearance of various possible combinations for these quantities one example given special attention is the limit set associated to the complex continued fraction expansion in particular lower and upper estimates for its hausdorff dimension are given a large natural class of systems whose limit sets are dimensionless in the restricted sense is described

elliptic quantum many body problem and double affine knizhnik zamolodchikov equation the elliptic matrix quantum olshanetsky perelomov problem is introduced for arbitrary root systems by means of an elliptic version of the dunkl operators its equivalence with the double affine generalization of the knizhnik zamolodchikov equation in the induced representations is established

algebraic linearization of dynamics of calogero type for any coxeter group calogero moser systems can be generalized for any root system including the noncrystallographic cases the algebraic linearization of the generalized calogero moser systems and of their quadratic respectively quartic perturbations are discussed

relativistic toda systems we present and study poincare invariant generalizations of the galilei invariant toda systems the classical nonperiodic systems are solved by means of an explicit action angle transformation

the hard hexagon partition function for complex fugacity we study the analyticity of the partition function of the hard hexagon model in the complex fugacity plane by computing zeros and transfer matrix eigenvalues for large finite size systems we find that the partition function per site computed by baxter in the thermodynamic limit for positive real values of the fugacity is not sufficient to describe the analyticity in the full complex fugacity plane we also obtain a new algebraic equation for the low density partition function per site