frozen glass phase in the multi index matching problem the multi index matching is an np hard combinatorial optimization problem for two indices it reduces to the well understood bipartite matching problem that belongs to the polynomial complexity class we use the cavity method to solve the thermodynamics of the multi index system with random costs the phase diagram is much richer than for the case of the bipartite matching problem it shows a finite temperature phase transition to a completely frozen glass phase similar to what happens in the random energy model we derive the critical temperature the ground state energy density and properties of the energy landscape and compare the results to numerics based on exact analysis of small systems

quantization of models with non compact quantum group symmetry modular xxz magnet and lattice sinh gordon model we define and study certain integrable lattice models with non compact quantum group symmetry the modular double of including an integrable lattice regularization of the sinh gordon model and a non compact version of the xxz model their fundamental r matrices are constructed in terms of the non compact quantum dilogarithm our choice of the quantum group representations naturally ensures self adjointness of the hamiltonian and the higher integrals of motion these models are studied with the help of the separation of variables method we show that the spectral problem for the integrals of motion can be reformulated as the problem to determine a subset among the solutions to certain finite difference equations baxter equation and quantum wronskian equation which is characterized by suitable analytic and asymptotic properties a key technical tool is the so called operator for which we give an explicit construction our results allow us to establish some connections to related results and conjectures on the sinh gordon theory in continuous spacetime our approach also sheds some light on the relations between massive and massless models in particular the sinh gordon and liouville theories from the point of view of their integrable structures

q bosons toda lattice pieri rules and baxter q operator we use the pieri rules to recover the q boson model and show it is equivalent to a discretized version of the relativistic toda chain we identify its semi infinite transfer matrix and the corresponding baxter q matrix with half vertex operators related by an duality transformation we observe that the scalar product of two higher spin xxz wave functions can be expressed with a gaudin determinant

on quantum separation of variables we present a new approach to construct the separate variables basis leading to the full characterization of the transfer matrix spectrum of quantum integrable lattice models the basis is generated by the repeated action of the transfer matrix itself on a generically chosen state of the hilbert space the fusion relations for the transfer matrix stemming from the yang baxter algebra properties provide the necessary closure relations to define the action of the transfer matrix on such a basis in terms of elementary local shifts leading to a separate transfer matrix spectral problem hence our scheme extends to the quantum case a key feature of the liouville arnold classical integrability framework where the complete set of conserved charges defines both the level manifold and the flows on it leading to the construction of action angle variables we work in the framework of the quantum inverse scattering method as a first example of our approach we give the construction of such a basis for models associated with y gln and argue how it extends to their trigonometric and elliptic versions then we show how our general scheme applies concretely to fundamental models associated with the y gl2 and y gl3 r matrices leading to the full characterization of their spectrum for y gl2 and its trigonometric deformation a particular case of our method reproduces sklyanin s construction of separate variables for y gl3 it gives new results in particular through the proper identification of the shifts acting on the separate basis we stress that our method also leads to the full characterization of the spectrum of other known quantum integrable lattice models including in particular trigonometric and elliptic spin chains open chains with general integrable boundaries and further higher rank cases that we will describe in forthcoming publications

chiral potts model as a descendant of the six vertex model we observe that then state integrable chiral potts model can be considered as a part of some new algebraic structure related to the six vertex model as a result we obtain a functional equation which is supposed to determine all the eigenvalues of the chiral potts model transfer matrix

sov approach for integrable quantum models associated with general representations on spin 1 2 chains of the 8 vertex reflection algebra the analysis of the transfer matrices associated with the most general representations of the 8 vertex reflection algebra on spin 1 2 chains is here implemented by introducing a quantum separation of variables sov method which generalizes to these integrable quantum models the method first introduced by sklyanin for representations reproducing in their homogeneous limits the open xyz spin 1 2 quantum chains with the most general integrable boundary conditions we explicitly construct representations of the 8 vertex reflection algebras for which the transfer matrix spectral problem is separated then in these sov representations we get the complete characterization of the transfer matrix spectrum eigenvalues and eigenstates and its non degeneracy moreover we present the first fundamental step toward the characterization of the dynamics of these models by deriving determinant formulae for the matrix elements of the identity on separated states which particularly apply to transfer matrix eigenstates a comparison of our analysis of the 8 vertex reflection algebra with that of niccoli g 2012 j stat mech p10025 faldella s et al 2014 j stat mech p01011 for the 6 vertex leads to an interesting remark in that there is a profound similarity in both the characterization of the spectral problems and the scalar products which exists for these two different realizations of the reflection algebra once they are described by the sov method as will be shown in a future publication this remarkable similarity will be the basis of a simultaneous determination of the form factors of local operators of integrable quantum models associated with general reflection algebra representations of both 8 vertex and 6 vertex type

the core of games on convex geometries abstract a game on a convex geometry is a real valued function defined on the family l of the closed sets of a closure operator which satisfies the finite minkowski krein milman property if l is the boolean algebra 2n then we obtain a n person cooperative game we will introduce convex and quasi convex games on convex geometries and we will study some properties of the core and the weber set of these games

the general uq sl 2 invariant xxz integrable quantum spin chain the boundary conditions are constructed for the general xxz quantum integrable spin chain such that the resulting system is invariant under the quantum algebra uq sl 2 action the result generalizes that due to pasquier and saleur 1990 obtained for spin 1 2 and spin 1

classical limit of the three point function of n 4 supersymmetric yang mills theory from integrability we give an analytic expression for the correlation function of three large classical nonprotected operators in n 4 super yang mills theory at weak coupling we restrict ourselves to operators belonging to an su 2 sector of the theory in order to carry out the calculation we derive by unveiling a hidden factorization property the thermodynamical limit of slavnov s determinant

slavnov and gaudin korepin formulas for models without u 1 symmetry the xxx chain on the segment we consider the isotropic spin heisenberg chain with the most general integrable boundaries the scalar product between the on shell bethe vector and its off shell dual obtained by means of the modified algebraic bethe ansatz is given by a modified slavnov formula the corresponding gaudin korepin formula i e the square of the norm is also obtained

structure of matrix elements in the quantum toda chain we consider the quantum toda chain using the method of separation of variables we show that the matrix elements of operators in the model are written in terms of a finite number of deformed abelian integrals the properties of these integrals are discussed we explain that these properties are necessary in order to provide the correct number of independent operators a comparison with the classical theory is made

the marriage problem from the bar of appointments to the agency abstract we study the stable marriage problem from different points of view we proposed a microscopic dynamic that led the system to a stationary state that we are able to characterize analytically then we derive a thermodynamical description of the nash equilibrium states of the system that agree very well with the results of monte carlo simulations finally through large scale numerical simulations we compare the global optimum of the society with the stable marriage of lower energy we showed that both states are strongly correlated and that the selfish attitude results in a benefit for most of the practitioners belonging to blocking pairs in the global optimum of the society

matrix coordinate bethe ansatz applications to xxz and asep models we present the construction of the full set of eigenvectors of the open asymmetric simple exclusion process asep and xxz models with special constraints on the boundaries the method combines both recent constructions of coordinate bethe ansatz and the old method of matrix ansatz specific to the asep this matrix coordinate bethe ansatz can be viewed as a non commutative coordinate bethe ansatz the non commutative part being related to the algebra appearing in the matrix ansatz

statistical mechanics of a simplified bipartite matching problem an analytical treatment we perform an analytical study of a simplified bipartite matching problem in which there exists a constant matching energy and both heterosexual and homosexual pairings are allowed we obtain the partition function in a closed analytical form and we calculate the corresponding thermodynamic functions of this model we conclude that the model is favored at high temperatures for which the probabilities of heterosexual and homosexual pairs tend to become equal in the limits of low and high temperatures the system is extensive however this property is lost in the general case there exists a relation between the matching energies for which the system becomes more stable under external thermal perturbations as the difference of energies between the two possible matches increases the system becomes more ordered while the maximum of entropy is achieved when these energies are equal in this limit there is a first order phase transition between two phases with constant entropy

boundary conditions for integrable quantum systems a new class of boundary conditions is described for quantum systems integrable by means of the quantum inverse scattering r matrix method the method proposed allows the author to treat open quantum chains with appropriate boundary terms in the hamiltonian the general considerations are applied to the xxz and xyz models the nonlinear schrodinger equation and toda chain

semi classical analysis of the inner product of bethe states we study the inner product of two bethe states one of which is taken on shell in an inhomogeneous xxx chain in the sutherland limit where the number of magnons is comparable with the length l of the chain and the magnon rapidities arrange in a small number of macroscopically large bethe strings the leading order in the large l limit is known to be expressed through a contour integral of a dilogarithm here we derive the sub leading term our analysis is based on a new contour integral representation of the inner product in terms of a fredholm determinant we give two derivations of the sub leading term besides a direct derivation by solving a riemann hilbert problem we give a less rigorous but more intuitive derivation by field theoretical methods for that we represent the fredholm determinant as an expectation value in a fock space of chiral fermions and then bosonize we construct a collective field for the bosonized theory the short wave length part of which may be evaluated exactly while the long wave length part is amenable to a 1 l expansion our treatment thus results in a systematic 1 l expansion of structure factors within the sutherland limit

the marriage problem and the fate of bachelors abstract in the marriage problem a variant of the bi parted matching problem each member has a wish list expressing his her preference for all possible partners this list consists of random positive real numbers drawn from a certain distribution one searches the lowest cost for the society at the risk of breaking up pairs in the course of time minimization of a global cost function hamiltonian is performed with statistical mechanics techniques at a finite fictitious temperature the problem is generalized to include bachelors needed in particular when the groups have different size and polygamy exact solutions are found for the optimal solution t 0 the entropy is found to vanish quadratically in t also other evidence is found that the replica symmetric solution is exact implying at most a polynomial degeneracy of the optimal solution whether bachelors occur or not depends not only on their intrinsic qualities or lack thereof but also on global aspects of the chance for pair formation in society

the baxter bazhanov stroganov model separation of variables and the baxter equation the baxter bazhanov stroganov model also known as the 2 model has attracted much interest because it provides a tool for solving the integrable chiral potts model it can be formulated as a face spin model or via cyclic l operators using the latter formulation and the sklyanin kharchev lebedev approach we give the explicit derivation of the eigenvectors of the component bn of the monodromy matrix for the fully inhomogeneous chain of finite length for the periodic chain we obtain the baxter t q equations via separation of variables the functional relations for the transfer matrices of the 2 model guarantee nontrivial solutions to the baxter equations for the n 2 case which is the free fermion point of a generalized ising model the baxter equations are solved explicitly

completeness of the bethe ansatz solution of the open xxz chain with nondiagonal boundary terms a bethe ansatz solution of the open spin xxz quantum spin chain with nondiagonal boundary terms has recently been proposed using a numerical procedure developed by mccoy et al we find significant evidence that this solution can yield the complete set of eigenvalues for generic values of the bulk and boundary parameters satisfying one linear relation moreover our results suggest that this solution is practical for investigating the ground state of this model in the thermodynamic limit

the open xxx spin chain in the sov framework scalar product of separate states in our previous paper kitanine et al 2017 j phys a math theor 50 224001 we have obtained for the xxx spin 1 2 heisenberg open chain new determinant representations for the scalar products of separate states in the quantum separation of variables sov framework in this article we perform a similar study in a more complicated case the xxz open spin 1 2 chain with the most general integrable boundary terms we first revisit the sov solution of this model using a matrix version of the vertex irf gauge transformation so as to reduce one of the boundary k matrices to a diagonal form as usual within the sov approach the scalar products of separate states are computed in terms of dressed vandermonde determinants having an intricate dependency on the inhomogeneity parameters we show that these determinants can be transformed into different ones in which the homogeneous limit can be taken straightforwardly these representations generalize in a non trivial manner to the trigonometric case the expressions found previously in the rational case we also show that generically all scalar products can be expressed in a form which is similar to although more cumbersome than the well known slavnov determinant representation for the scalar products of the bethe states of the periodic chain considering a special choice of the boundary parameters relevant in the thermodynamic limit to describe the half infinite chain with a general boundary we particularize these representations to the case of one of the two states being an eigenstate we obtain simplified formulas that should be of direct use to compute the form factors and correlation functions of this model

q deformations of the o 3 symmetric spin 1 heisenberg chain the authors present the general expression for the spin 1 heisenberg chain invariant under the uq so 3 quantum algebra several physical and mathematical implications are discussed

a ranking model for the greedy algorithm and discrete convexity generalizing the idea of the lovasz extension of a set function and the discrete choquet integral we introduce a combinatorial model that allows us to define and analyze matroid type greedy algorithms the model is based on a real valued function v on a finite family of sets which yields the constraints of a combinatorial linear program moreover v gives rise to a ranking and selection procedure for the elements of the ground set n and thus implies a greedy algorithm for the linear program it is proved that the greedy algorithm is guaranteed to produce primal and dual optimal solutions if and only if an associated functional on mathbb r n is concave previous matroid type greedy models are shown to fit into the present general context in particular a general model for combinatorial optimization under supermodular constraints is presented which guarantees the greedy algorithm to work

analytic properties of thermodynamic functions at first order phase transitions develops a method of analytic continuation at first order phase transitions and applies it to the d 2 ising model in an external field the method employs a function built of transfer matrix eigenvalues which provides rapidly convergent approximations to the stable free energy f and its derivatives for all h or 0 recent series analysis results on the existence of an essential singularity at h 0 are confirmed there is also an indication of a spinodal line and hsp t o is estimated in the range o t or 0 6tc

form factors in the baxter bazhanov stroganov model i norms and matrix elements we continue our investigation of the baxter bazhanov stroganov model using the method of separation of variables 1 in this paper we calculate the norms and matrix elements of a local spin operator between eigenvectors of the auxiliary problem for the norm the multiple sums over the intermediate states are performed explicitly in the case n 2 we solve the baxter equation and obtain form factors of the spin operator of the periodic ising model on a finite lattice

bethe ansatz solution of the open xxz chain with nondiagonal boundary terms we consider the integrable open xx quantum spin chain with non diagonal boundary terms we derive an exact inversion identity by which we obtain the eigenvalues of the transfer matrix and the bethe ansatz equations for generic values of the boundary parameters the bethe ansatz solution is formulated in terms of the jacobian elliptic functions

boundary k matrices for the six vertex and the n 2n 1 an 1 vertex models boundary conditions compatible with integrability are obtained for two dimensional models by solving the factorizability equations for the reflection matrices k or theta for the six vertex model the general solution depending on four arbitrary parameters is found for the an 1 models all diagonal solutions are found the associated integrable magnetic hamiltonians are explicitly derived

determinant formula for the six vertex model with reflecting end using the quantum inverse scattering method for the xxz model with open boundary conditions we obtained the determinant formula for the six vertex model with reflecting end

form factors and complete spectrum of xxx antiperiodic higher spin chains by quantum separation of variables the antiperiodic transfer matrices associated to higher spin representations of the rational 6 vertex yang baxter algebra are analyzed by generalizing the approach introduced recently in the framework of sklyanin s quantum separation of variables sov for cyclic representations spin 1 2 highest weight representations and also for spin 1 2 representations of the 6 vertex reflection algebra such sov approach allow us to derive exactly results which represent complicate tasks for more traditional methods based on bethe ansatz and baxter q operator in particular we both prove the completeness of the sov characterization of the transfer matrix spectrum and its simplicity then the derived characterization of local operators by sklyanin s quantum separate variables and the expression of the scalar products of separate states by determinant formulae allow us to compute the form factors of the local spin operators by one determinant formulae similar to those of the scalar products

on the restricted cores and the bounded core of games on distributive lattices a game with precedence constraints is a tu game with restricted cooperation where the set of feasible coalitions is a distributive lattice hence generated by a partial order on the set of players its core may be unbounded and the bounded core which is the union of all bounded faces of the core proves to be a useful solution concept in the framework of games with precedence constraints replacing the inequalities that define the core by equations for a collection of coalitions results in a face of the core a collection of coalitions is called normal if its resulting face is bounded the bounded core is the union of all faces corresponding to minimal normal collections we show that two faces corresponding to distinct normal collections may be distinct moreover we prove that for superadditive games and convex games only intersecting and nested minimal collection respectively are necessary finally it is shown that the faces corresponding to pairwise distinct nested normal collections may be pairwise distinct and we provide a means to generate all such collections

an antiperiodic dynamical six vertex model i complete spectrum by sov matrix elements of the identity on separate states and connections to the periodic eight vertex model the spin 1 2 highest weight representations of the dynamical six vertex and the standard eight vertex yang baxter algebra on a finite chain are considered in this paper in particular the integrable quantum models associated with the corresponding transfer matrices under antiperiodic boundary conditions for the dynamical six vertex case and periodic boundary conditions for the eightvertexcaseareanalyzedhere fortheantiperiodicdynamicalsix vertextransfer matrix defined on chains with an odd number of sites we adapt sklyanin s quantum separation of variable sov method and explicitly construct the sov representations from the original space of the representations in this way we provide the complete characterization of the eigenvalues and the eigenstates proving also the simplicity of its spectrum moreover we characterize the matrix elements of the identity on separated states of this model by determinant formulae the matrices entering these determinants have elements given by sums over the sov spectrum of the product of the coefficients of the separate states this sov analysis is done without any need to be reduced to the case of the so called elliptic roots of unit and the results derived here define the required setup to extend to the dynamical six vertex model the approach recently developed by the author and collaborators to compute the form factors of the local operators in the sov framework for the periodic eight vertex transfer matrix we prove that its eigenvalues have to satisfy a fixed system of equations in the case of a chain with an odd number of sites this system of equations is the same entering in the sov characterization of the antiperiodic dynamical six vertex transfer matrix spectrum this implies that the set of the periodic eight vertex eigenvalues is contained in the set of the antiperiodic dynamical six vertex eigenvalues a criterion is introduced to find simultaneous eigenvalues of these two transfer matrices and associate with any of such eigenvalues one nonzero eigenstate of the periodic eight vertex transfer matrix by using the sov results moreover a preliminary discussion

on determinant representations of scalar products and form factors in the sov approach the xxx case in the present article we study the form factors of quantum integrable lattice models solvable by the separation of variables sovs method it was recently shown that these models admit universal determinant representations for the scalar products of the so called separate states a class which includes in particular all the eigenstates of the transfer matrix these results permit to obtain simple expressions for the matrix elements of local operators form factors however these representations have been obtained up to now only for the completely inhomogeneous versions of the lattice models considered in this article we give a simple algebraic procedure to rewrite the scalar products and hence the form factors for the sov related models as izergin or slavnov type determinants this new form leads to simple expressions for the form factors in the homogeneous and thermodynamic limits to make the presentation of our method clear we have chosen to explain it first for the simple case of the xxx heisenberg chain with anti periodic boundary conditions we would nevertheless like to stress that the approach presented in this article applies as well to a wide range of models solved in the sov framework

monge extensions of cooperation and communication structures abstract cooperation structures without any a priori assumptions on the combinatorial structure of feasible coalitions are studied and a general theory for marginal values cores and convexity is established the theory is based on the notion of a monge extension of a general characteristic function which is equivalent to the lovasz extension in the special situation of a classical cooperative game it is shown that convexity of a cooperation structure is tantamount to the equality of the associated core and weber set extending myerson s graph model for game theoretic communication general communication structures are introduced and it is shown that a notion of supermodularity exists for this class that characterizes convexity and properly extends shapley s convexity model for classical cooperative games

liouville theory ads _2 string and three point functions this is a write up of the lectures given in young researchers integrability school 2017 a 13 the main goal is to explain the connection between the ode im correspondence and thea 13 classical integrability of strings in ads as a warm up we first discuss the classical three pointa 13 function of the liouville theory our starting point is the well known fact that the classicala 13 solutions to the liouville equation can be constructed by solving a certain schrodinger likea 13 differential equation we then convert it into a set of functional equations using the ideaa 13 akin to the ode im correspondence the classical three point functions can be computeda 13 directly from these functional equations and the result matches with the classical limit ofa 13 the celebrated dozz formula we then proceed to discuss the semi classical three pointa 13 function of strings in ads2 and show that one can apply a similar idea by making use of thea 13 classical integrability of the string sigma model on ads2

quantization of solitons and the restricted sine gordon model we show how to compute form factors matrix elements of local fields in the restricted sine gordon model at the reflectionless points by quantizing solitons we introduce quantum separated variables in which the hamiltonians are expressed in terms of quantum functions we explicitly describe the soliton wave functions and we explain how the restriction is related to an unusual hermitian structure we also present a semi classical analysis which enlightens the fact that the restricted sine gordon model corresponds to an analytical continuation of the sine gordon model intermediate between sine gordon and kdv

boundary k matrices for the xyz xxz and xxx spin chains the general solutions for the factorization equations of the reflection matrices k or theta for the eight vertex and six vertex models xyz xxz and xxx chains are found the associated integrable magnetic hamiltonians are derived explicitly finding families depending on several continuous as well as discrete parameters

null vectors in integrable field theory the form factor bootstrap approach allows to construct the space of local fields in the massive restricted sine gordon model this space has to be isomorphic to that of the corresponding minimal model of conformal field theory we describe the subspaces which correspond to the verma modules of primary fields in terms of the commutative algebra of local integrals of motion and of a fermion neveu schwarz or ramond depending on the particular primary field the description of null vectors relies on the relation between form factors and deformed hyper elliptic integrals the null vectors correspond to the deformed exact forms and to the deformed riemann bilinear identity in the operator language the null vectors are created by the action of two operators linear in the fermion and quadratic in the fermion we show that by factorizing out the null vectors one gets the space of operators with the correct character in the classical limit using the operators and we obtain a new very compact description of the kdv hierarchy we also discuss a beautiful relation with the method of whitham

on the core of ordered submodular cost games a general ordertheoretic linear programming model for the study of matroid type greedy algorithms is introduced the primal restrictions are given by so called weakly increasing submodular functions on antichains the lp dual is solved by a monge type greedy algorithm the model offers a direct combinatorial explanation for many integrality results in discrete optimization in particular the submodular intersection theorem of edmonds and giles is seen to extend to the case with a rooted forest as underlying structure the core of associated polyhedra is introduced and applications to the existence of the core in cooperative game theory are discussed

sex oriented stable matchings of the marriage problem with correlated and incomplete information in the stable marriage problem two sets of agents must be paired according to mutual preferences which may happen to conflict we present two generalizations of its sex oriented version aiming to take into account correlations between the preferences of agents and costly information their effects are investigated both numerically and analytically