the replica symmetric solution for potts models on d regular graphs we establish an explicit formula for the limiting free energy density log partition function divided by the number of vertices for ferromagnetic potts models on uniformly sparse graph sequences converging locally to the d regular tree for d even covering all temperature regimes this formula coincides with the bethe free energy functional evaluated at a suitable fixed point of the belief propagation recursion on the d regular tree the so called replica symmetric solution for uniformly random d regular graphs we further show that the replica symmetric bethe formula is an upper bound for the asymptotic free energy for any model with permissive interactions

the order parameter in a spin glass various possible precise definitions of an edwards anderson type of order parameter for an ising model spin glass are considered using boundary conditions for a finite system states of an infinite system and a duplicate system approach several of these definitions are shown to yield identical results

poisson hopf limit of quantum algebras the poisson hopf analogue of an arbitrary quantum algebra uz g is constructed by introducing a one parameter family of quantizations uz g depending explicitly on and by taking the appropriate 0 limit the q poisson analogues of the su 2 algebra are discussed and the novel case is introduced the q serre relations are also extended to the poisson limit this approach opens the perspective for possible applications of higher rank q deformed hopf algebras in semiclassical contexts

biorthogonal coupling coefficients of uq n the coupling wigner clebsch gordan coefficients and their isofactors for the unitary quantum algebras uq n with the repeating irreducible representations in the coproduct decomposition are considered generalizing the u n case the biorthogonal systems of the uq n isofactors with the dual multiplicity labels are constructed by means of the recoupling technique in terms of the isofactors with simpler multiplicity structure a first construction correlated with the inverted littlewood richardson rules gives the bilinear combinations of isofactors after applying the proportionality of the q recoupling racah coefficients to the boundary q isofactors an alternative recursive construction gives the nonorthogonal q isofactors satisfying the most elementary boundary conditions and proportional to the uq n 1 recoupling coefficients for some less restricted values of parameters some multiplicity free and more general uq n recoupling coefficients are found the blocks bilinear combinations of which equal to the resubducing coefficients of the complementary chains of q algebras are proposed to use for the orthonormalization of some uq n biorthogonal isofactors including the general uq 3 case

the drinfeld double gl n tn we present a self dual drinfeld double structure underlying the an series of simple lie algebras such double is constructed through a central extension tn of gl n and is obtained by pairing two disjoint solvable subalgebras coming from positive and negative roots the cartan weyl basis of gl n is shown to be completely determined by the compatibility conditions in the double a natural lie bialgebra structure on gl n is obtained that offers a new perspective for the construction of its quantum deformations

multiplicity free uq n coupling coefficients the coupling wigner clebsch gordan coefficients of the unitary quantum algebras uq n are considered the tensorial properties of the generator powers and their ordered products used in the explicit projectors and weight lowering procedures are established different expressions for the multiplicity free isofactors of uq n coupling coefficients those coupling an arbitrary and symmetric representations are derived explicit expression of the arbitrary isofactors in terms of their boundary values are proposed proportionality of the semistretched isofactors to the stretched q recoupling coefficients of uq n 1 q analogues of 9j symbols is demonstrated the stretched isofactors of uq 3 are expressed in terms of clebsch gordan coefficients of uq 2

bethe lattice spin glass the effects of a ferromagnetic bias and external fields i bifurcation analysis we present a rigorous analysis of the j ising spin glass model on the bethe lattice with fixed uncorrelated boundary conditions phase diagrams are derived as a function of temperature vs concentration of ferromagnetic bonds and for a symmetric distribution of bonds external field vs temperature in this part we characterize the bulk ordered phases using bifurcation theory we prove the existence of a distribution of single site magnetizations far inside the lattice which is stable with respect to changes in the boundary conditions

classical lie algebras and drinfeld doubles the drinfeld double structure underlying the cartan series an bn cn dn of simple lie algebras is discussed this structure is determined by two disjoint solvable subalgebras matched by a pairing for the two nilpotent positive and negative root subalgebras the pairing is natural and in the cartan subalgebra is defined with the help of a central extension of the algebra a new completely determined basis is found from the compatibility conditions in the double and a different perspective for quantization is presented other related drinfeld doubles on are also considered

the tightness of the kesten stigum reconstruction bound of symmetric model with multiple mutations it is well known that reconstruction problems as the interdisciplinary subject have been studied in numerous contexts including statistical physics information theory and computational biology to name a few we consider a 2q state symmetric model with two categories of q states in each category and 3 transition probabilities the probability to remain in the same state the probability to change states but remain in the same category and the probability to change categories we construct a nonlinear second order dynamical system based on this model and show that the kesten stigum reconstruction bound is not tight when q ge 4

a mean field spin glass with short range interactions we formulate and study a spin glass model on the bethe lattice appropriate boundary fields replace the traditional self consistent methods they give our model well defined thermodynamic properties we establish that there is a spin glass transition temperature above which the single site magnetizations vanish and below which the edwards anderson order parameter is strictly positive in a neighborhood below the transition temperature we use bifurcation theory to establish the existence of a nontrivial distribution of single site magnetizations two properties of this distribution are studied the leading perturbative correction to the gaussian scaling form at the transition and the nonperturbative behavior of the tails

the 2 category theory of quasi categories abstract in this paper we re develop the foundations of the category theory of quasi categories also called categories using 2 category theory we show that joyal s strict 2 category of quasi categories admits certain weak 2 limits among them weak comma objects we use these comma quasi categories to encode universal properties relevant to limits colimits and adjunctions and prove the expected theorems relating these notions these universal properties have an alternate form as absolute lifting diagrams in the 2 category which we show are determined pointwise by the existence of certain initial or terminal vertices allowing for the easy production of examples all the quasi categorical notions introduced here are equivalent to the established ones but our proofs are independent and more formal in particular these results generalise immediately to model categories enriched over quasi categories

graph partitioning and dilute spin glasses the minimum cost solution the authors calculate the ground state cost function energy for graph bipartitioning and spin glasses on networks with fixed finite valences allowing a continuous distribution for the effective field compared with previous calculations using a field distribution in integral multiples of the coupling strength j the ground state energy is higher by less than one per cent although the distribution itself is altered drastically

wishart planted ensemble a tunably rugged pairwise ising model with a first order phase transition we propose the wishart planted ensemble a class of zero field ising models with tunable algorithmic hardness and specifiable or planted ground state the problem class arises from a simple procedure for generating a family of random integer programming problems with specific statistical symmetry properties but turns out to have intimate connections to a sign inverted variant of the hopfield model the hamiltonian contains only 2 spin interactions with the coupler matrix following a type of wishart distribution the class exhibits a classical first order phase transition in temperature for some parameter settings the model has a locally stable paramagnetic state a feature which correlates strongly with difficulty in finding the ground state and suggests an extremely rugged energy landscape we analytically probe the ensemble thermodynamic properties by deriving the thouless anderson palmer equations and free energy and corroborate the results with a replica and annealed approximation analysis extensive monte carlo simulations confirm our predictions of the first order transition temperature the class exhibits a wide variation in algorithmic hardness as a generation parameter is varied with a pronounced easy hard easy profile and peak in solution time towering many orders of magnitude over that of the easy regimes by deriving the ensemble averaged energy distribution and taking into account finite precision representation we propose an analytical expression for the location of the hardness peak and show that at fixed precision the number of constraints in the integer program must increase with system size to yield truly hard problems the wishart planted ensemble is interesting for its peculiar physical properties and provides a useful and analytically transparent set of problems for benchmarking optimization algorithms

from quantum universal enveloping algebras to quantum algebras the local structure of a quantum group gq is currently considered to be an infinite dimensional object the corresponding quantum universal enveloping algebra uq g which is a hopf algebra deformation of the universal enveloping algebra of an n dimensional lie algebra g lie g however we show how by starting from the generators of the underlying lie bialgebra g the analyticity in the deformation parameter s allows us to determine in a unique way a set of n almost primitive basic objects in uq g which could be properly called the quantum algebra generators so the analytical prolongation gq of the lie bialgebra g is proposed as the appropriate local structure of gq besides as in this way g and uq g are shown to be in one to one correspondence the classification of quantum groups is reduced to the classification of lie bialgebras the suq 2 and suq 3 cases are explicitly elaborated

a model category structure on the category of simplicial categories in this paper we put a cofibrantly generated model category struc ture on the category of small simplicial categories the weak equivalences are a simplicial analogue of the notion of equivalence of categories

the random cluster model on a homogeneous tree the random cluster model on a homogeneous tree is defined and studied it is shown that for 1 q 2 the percolation probability in the maximal random cluster measure is continuous inp while forq 2 it has a discontinuity at the critical valuep p c q it is also shown that forq 2 there is nonuniqueness of random cluster measures for an entire interval of values ofp the latter result is in sharp contrast to what happens on the integer lattice z d

a homotopy coherent cellular nerve for bicategories abstract the subject of this paper is a nerve construction for bicategories introduced by leinster which defines a fully faithful functor from the category of bicategories and normal pseudofunctors to the category of presheaves over joyal s category 2 we prove that the nerve of a bicategory is a 2 quasi category a model for 2 categories due to ara and moreover that the nerve functor restricts to the right part of a quillen equivalence between lack s model structure for bicategories and a bousfield localisation of ara s model structure for 2 quasi categories we deduce that lack s model structure for bicategories is quillen equivalent to rezk s model structure for 2 2 spaces on the category of simplicial presheaves over 2 to this end we construct the homotopy bicategory of a 2 quasi category and prove that a morphism of 2 quasi categories is an equivalence if and only if it is essentially surjective on objects and fully faithful we also prove a quillen equivalence between ara s model structure for 2 quasi categories and the hirschowitz simpson pellissier model structure for quasi category enriched segal categories from which we deduce a few more results about 2 quasi categories including a conjecture of ara concerning weak equivalences of 2 categories

three dimensional quantum algebras a cartan like point of view a perturbative quantization procedure for lie bialgebras is introduced the relevance of the choice of a completely symmetrized basis of the quantum universal enveloping algebra is stressed sets of elements of the quantum algebra that play a role similar to generators in the case of lie algebras are considered and a cartan like procedure applied to find a representative for each class of quantum algebras the method is used to construct and classify all three dimensional complex quantum algebras that are compatible with a given type of coproduct the quantization of all lie algebras that in the classical limit belong to the most relevant sector in the classification for three dimensional lie bialgebras given in 1 is thus performed new quantizations of solvable algebras whose simplicity makes them suitable for possible physical applications are obtained and already known related quantum algebras recovered

on mean field equations for spin glasses in this paper we study rigorously the random ising model on a cayley tree in the limit of infinite coordination numberz 8 an iterative scheme is developed relating mean magnetizations and mean square magnetizations of successive shells far removed from the surface of the lattice in this way we obtain local properties of the model in the thermodynamic limit of an infinite number of shells when the coupling constants are independent gaussian random variables the sk expressions emerge as stable fixed points of our scheme and provide a valid local mean field theory of spin glasses in which negative local entropy at low temperatures while perfectly possible mathematically may still perhaps be physically undesirable finally we examine the tap equations and show that if the average over bond disorder and the limitz 8 are actually performed one recovers our iterative scheme and hence the sk equations also in the thermodynamic limit

a conjectured scenario for order parameter fluctuations in spin glasses we study order parameter fluctuations opf in disordered systems by considering the behaviour of some recently introduced parameters g gc which have proven very useful in locating phase transitions we prove that both parameters g for disconnected overlap disorder averages and gc for connected disorder averages take the respective universal values 1 3 and 13 31 in the t 0 limit for any finite volume provided the ground state is unique and there is no gap in the ground state local field distributions conditions which are met in generic spin glass models with continuous couplings and no gap at zero coupling this makes g gc ideal parameters to locate phase transitions in disordered systems much like the binder cumulant for ordered systems we check our results by exactly computing opf in a simple example of uncoupled spins in the presence of random fields and the one dimensional ising spin glass at finite temperatures we discuss under which conditions the value 1 3 for g may be recovered by conjecturing different scenarios depending on whether opf are finite or vanish in the infinite volume limit in particular we discuss replica equivalence and its natural consequence lim v g v t 1 3 when opf are finite as an example of a model where opf vanish and replica equivalence does not give information about g we study the sherrington kirkpatrick spherical spin glass model by performing numerical simulations for small sizes again we find results compatible with g 1 3 in the spin glass phase

one dimensional ising spin glass model with long range interactions an ising chain with hamiltonian h j sigma i j epsilon ijsisj mod i j mod 1 sigma 2 is considered where the epsilon ij are independent random variables according to kotliar et al 1983 this model has a phrase transition with non mean field exponents when 1 3 sigma 1 and mean field exponents beyond the upper critical value of sigma 1 3 by means of an epsilon expansion about the lower critical value of sigma 1 the model is shown to be replica symmetric as epsilon to 0 and it is speculated that this result holds for 1 3 sigma 1

random solutions from a regular density functional hamiltonian a static and dynamical theory for the structural glass transition as simple model density functional hamiltonian is shown to lead to a consistent static and dynamical theory of the structural glass transition where glassy metastable free energy states play a key role the crucial concept introduced is a probabilistic order parameter description that results when multiple solutions of the density functional theory are considered where each solution is given a canonical weight the theory suggests that there are two distinct transition temperatures or densities at the higher transition temperature an extensive number of well defined global metastable states appear for the first time at the lower transition temperature the number of relevant glassy metastable states becomes non extensive

local properties of an ising model on a cayley tree the ising model on a cayley tree displays a peculiar continuous order phase transition with zero long range order at all finite temperatures when one studies expection values of spins far removed from the surface which contains a finite fraction of the total number of spins in the thermodynamic limit however one obtains the so called bethe approximation here we study such a local description by setting up a simple recurrence relation for successive shell magnetizations far removed from the surface in the ferromagnetic case the local magnetization is a fixed point of the iterative transformation while in the antiferromagnetic case the fixed point bifurcates to a two cycle of the transformation for low temperatures and fields giving rise to local sublattice magnetizations in both cases local thermodynamical properties are obtained by integration

on quantization of z2 graded algebras quantization of z2 graded lie bialgebras viewed as a generalization of sl 2 is investigated a method of building multiparameter solutions is developed we study the role of the quantum double construction in the selected class and prove that a quantized coboundary z2 graded lie bialgebra contains a quasitriangular hopf subalgebra

generalized quantization scheme for lie algebras the generalized form of the coproduct for quantum lie algebras is proposed it is demonstrated that the admissible quantum compositions can be easily found the whole construction gives rise to the wide class of multidimensional hopf algebras the efficiency of the method is demonstrated on examples of nonsemisimple quantum universal enveloping algebras

weighted averages and order parameters for the infinite range ising spin glass the sherrington kirkpatrick spin glass model 1975 is studied by replicas and by analysing the mean field equations of thouless anderson and palmer tap 1977 the authors show that the standard order parameter defined by statistical mechanics is given by the average of all off diagonal components of the matrix qalpha beta there is consequently no violation of the fluctuation dissipation theorem they describe how to weight the different solutions of the tap equations and argue that there is no entropy from solution degeneracy this assumption is shown to be internally consistent the square of a spin expectation value for a solution averaged over solutions is shown to be q 1 if they make the parisi sompolinsky ansatz for qalpha beta

inclusion of short range order in a mean field theory for the disordered magnetic lattice gas the disordered magnetic lattice gas dmlg as a unifying description of many simpler random spin systems has been investigated in an attempt to devise a mean field theory which goes beyond the infinitely long ranged model by incorporating short range order sro the authors have shown rigorously that the local thermodynamic properties of the dmlg on a cayley tree of finite coordination number z are identical to the thermodynamic properties of the dmlg in a pair approximation obtained by using the method of the distribution function further a modified pair approximation for the dmlg is presented which is exactly solvable it is formulated for general random bond distribution functions and is then examined for the special case of gaussian distributions

iterated spans and classical topological field theories we construct higher categories of iterated spans possibly equipped with extra structure in the form of higher categorical local systems and classify their fully dualizable objects by the cobordism hypothesis these give rise to framed topological quantum field theories which are the framed versions of the classical topological quantum field theories considered in the quantization programme of freed hopkins lurie teleman using this machinery we also construct an infty 1 category of symplectic derived algebraic stacks and lagrangian correspondences and show that all its objects are dualizable

mean field theory for percolation models of the ising type the q 2 random cluster model is studied in the context of two mean field models the bethe lattice and the complete graph for these systems the critical exponents that are defined in terms of finite clusters have some anomalous values as the critical point is approached from the high density side which vindicates the results of earlier studies in particular the exponent which characterizes the divergence of the average size of finite clusters is 1 2 and the exponent associated with the length scale of finite clusters is 1 4 the full collection of exponents indicates an upper critical dimension of 6 the standard mean field exponents of the ising system are also present in this model 1 2 1 which implies in particular the presence of two diverging length scales furthermore the finite cluster exponents are stable to the addition of disorder which near the upper critical dimension may have interesting implications concerning the generality of the disordered system correlation length bounds

generalized lie algebras of type an it is shown that the quantized enveloping algebra of sl n contains a generalized lie algebra defined by means of axioms similar to woronowicz s this gives rise to lie algebra like generators and relations for the locally finite part of the quantized enveloping algebra and suggests a canonical poincare birkhoff witt basis

zero temperature hysteresis in the random field ising model on a bethe lattice we consider the single spin flip dynamics of the random field ising model on a bethe lattice at zero temperature in the presence of a uniform external field we determine the average magnetization as the external field is varied from to by setting up the self consistent field equations which we show are exact in this case the qualitative behaviour of magnetization as a function of the external field unexpectedly depends on the coordination number z of the bethe lattice for z 3 with a gaussian distribution of the quenched random fields we find no jump in magnetization for any non zero strength of disorder for for weak disorder the magnetization shows a jump discontinuity as a function of the external uniform field which disappears for a larger variance of the quenched field we determine exactly the critical point separating smooth hysteresis curves from those with a jump we have checked our results by monte carlo simulations of the model on three and four coordinated random graphs which for large system sizes give the same results as on the bethe lattice but avoid surface effects altogether

the structure of quantum lie algebras for the classical series and the structure constants of quantum lie algebras depend on a quantum deformation parameter q and they reduce to the classical structure constants of a lie algebra at q 1 we explain the relationship between the structure constants of quantum lie algebras and quantum clebsch gordan coefficients for we present a practical method for the determination of these quantum clebsch gordan coefficients and are thus able to give explicit expressions for the structure constants of the quantum lie algebras associated to the classical lie algebras and in the quantum case the structure constants of the cartan subalgebra are non zero and we observe that they are determined in terms of the simple quantum roots we introduce an invariant killing form on the quantum lie algebras and find that it takes values which are simple q deformations of the classical ones

graph bipartitioning and the bethe spin glass the problem of the equipartitioning of a random graph of fixed finite valence is studied by comparison with a ferromagnetic bethe lattice with random boundary conditions the simplicity of recursion relations for effective fields due to descendents on bethe lattices provides simple approximations for the optimal cost in quite good agreement with simulations

homotopy coherent adjunctions and the formal theory of monads abstract in this paper we introduce a cofibrant simplicial category that we call the free homotopy coherent adjunction and characterise its n arrows using a graphical calculus that we develop here the hom spaces are appropriately fibrant indeed are nerves of categories which indicates that all of the expected coherence equations in each dimension are present to justify our terminology we prove that any adjunction of quasi categories extends to a homotopy coherent adjunction and furthermore that these extensions are homotopically unique in the sense that the relevant spaces of extensions are contractible kan complexes we extract several simplicial functors from the free homotopy coherent adjunction and show that quasi categories are closed under weighted limits with these weights these weighted limits are used to define the homotopy coherent monadic adjunction associated to a homotopy coherent monad we show that each vertex in the quasi category of algebras for a homotopy coherent monad is a codescent object of a canonical diagram of free algebras to conclude we prove the quasi categorical monadicity theorem describing conditions under which the canonical comparison functor from a homotopy coherent adjunction to the associated monadic adjunction is an equivalence of quasi categories our proofs reveal that a mild variant of beck s argument is all in the weights much of it independent of the quasi categorical context

quantization of drinfel d doubles hopf algebra quantizations of four dimensional and six dimensional real classical drinfel d doubles are studied by following a direct analytic approach and the full quantization is explicitly obtained for most of them several new four and six dimensional quantum algebras are presented and some general features of the method are emphasized

on quantum lie algebras and quantum root systems as a natural generalization of ordinary lie algebras we introduce the concept of quantum lie algebras we define these in terms of certain adjoint submodules of quantized enveloping algebras endowed with a quantum lie bracket given by the quantum adjoint action the structure constants of these algebras depend on the quantum deformation parameter q and they go over into the usual lie algebras when q 1 the notions of q conjugation and q linearity are introduced q linear analogues of the classical antipode and cartan involution are defined and a generalized killing form q linear in the first entry and linear in the second is obtained these structures allow the derivation of symmetries between the structure constants of quantum lie algebras the explicitly worked out examples of and illustrate the results

the galois group of a stable homotopy theory abstract to a stable homotopy theory a presentable symmetric monoidal stable category we naturally associate a category of finite etale algebra objects and using grothendieck s categorical machine a profinite group that we call the galois group we then calculate the galois groups in several examples for instance we show that the galois group of the periodic e algebra of topological modular forms is trivial and that the galois group of k n local stable homotopy theory is an extended version of the morava stabilizer group we also describe the galois group of the stable module category of a finite group a fundamental idea throughout is the purely categorical notion of a descendable algebra object and an associated analog of faithfully flat descent in this context

a remark on the notion of robust phase transitions we point out that the high qpotts model on a regular lattice at its transition temperature provides an example of a nonrobust in the sense recently proposed by pemantle and steif phase transition

derived koszul duality and tq homology completion of structured ring spectra abstract working in the context of symmetric spectra we consider algebraic structures that can be described as algebras over an operad o topological quillen homology or tq homology is the spectral algebra analog of quillen homology and stabilization working in a framework for homotopical descent the associated tq homology completion map appears as the unit map of an adjunction comparing the homotopy categories of o algebra spectra with coalgebra spectra over the associated comonad via tq homology we prove that this adjunction between homotopy categories can be turned into an equivalence by replacing o algebras with the full subcategory of 0 connected o algebras we also construct the spectral algebra analog of the unstable adams spectral sequence that starts from the tq homology groups tq x of an o algebra x and prove that it converges strongly to x when x is 0 connected

stiffness exponent number of pure states and almeida thouless line in spin glasses we show how an inequality for the stiffness exponent for spin glass models proposed by fisher and huse could be violated we analyze their derivation and point out that their scaling arguments do not apply to investigations of the difference between systems with periodic and antiperiodic boundary conditions as a consequence the possibility remains open that in sufficiently high dimensions an infinite number of pure states exists and that an almeida thouless line spin glass transition in a field occurs as is predicted in competing theories