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

mappin class group actions on quantum doubles we study representations of the mapping class group of the punctured torus on the double of a finite dimensional possibly non semisimple hopf algebra that arise in the construction of universal extended topological field theories we discuss how for doubles the degeneracy problem of tqft s is circumvented we find compact formulae for thes 1 matrices using the canonical non degenerate forms of hopf algebras and the bicrossed structure of doubles rather than monodromy matrices a rigorous proof of the modular relations and the computation of the projective phases is supplied using radford s relations between the canonical forms and the moduli of integrals we analyze the projectivesl 2 z action on the center ofuq sl2 forq anl 2m 1st root of unity it appears that the 3m 1 dimensional representation decomposes into anm 1 dimensional finite representation and a2m dimensional irreducible representation the latter is the tensor product of the two dimensional standard representation ofsl 2 z and the finite m dimensional representation obtained from the truncated tqft of the semisimplified representation category ofuq sl2

revisiting the cavity method threshold for random 3 sat a detailed monte carlo study of the satisfiability threshold for random 3 sat has been undertaken in combination with a monotonicity assumption we find that the threshold for random 3 sat satisfie

similarity analysis of inelastic contact analysis of mechanical contact of solids is of interest not only regarding a variety of mechanical assemblies but also on a smaller scale such as roughness properties of surfaces and compaction of powder particles indentation testing is another prominent problem in the context to analyse the phenomena involved is inherently difficult at application essentially due to the presence of large strains nonlinear material behaviour time dependence and moving contact boundaries recently progress has been made however to explicitly solve basic boundary value problems especially due to advances in computational techniques a substantial ingredient which facilitates solution procedures is self similarity and it is the present purpose to explore in detail the advantages in a general setting when this feature prevails a viscoplastic framework is laid down for a wide class of constitutive properties where strain hardening plasticity creep and also nonlinear elasticity arise as special cases it is then shown that when surface shapes and material properties are modelled by homogeneous functions associated boundary value problems posed may be reduced to stationary ones as a consequence within hertzian kinematics relations between contact impression and regions become independent of loading and time and the connection to loading characteristics does not usually require a full solution of the problem in particular it is shown that for general head shapes it proves efficient to use an approach where an intermediate flat die solution serves as a basic tool also for hereditary materials an invariant computational procedure based on the intermediate problem is arrived at and decisive results shown to be found by simple cumulative superposition illustrations are given analytically for ellipsoidal contact of newtonian fluids and by detailed computations for spherical indentation of viscoplastic solids for which also universal hardness formulae are proposed for several bodies in contact it is shown how general results may be extracted from fundamental solutions for a half space

integrality for tqfts we discuss ways that the ring of coefficients for a tqft can be reduced if one restricts somewhat the allowed cobordisms when we apply these methods to a tqft associated to so 3 at an odd prime p we obtain a functor from a somewhat restricted cobordism category to the category of free finitely generated modules over a ring of cyclotomic integers z 2p if p 1 mod 4 and z 4p if p 1 mod 4 where k is a primitive kth root of unity we study the quantum invariants of prime power order simple cyclic covers of 3 manifolds we define new invariants arising from strong shift equivalence and integrality similar results are obtained for some other tqfts but the modules are only guaranteed to be projective

improved belief propagation algorithm finds many bethe states in the random field ising model on random graphs we first present an empirical study of the belief propagation bp algorithm when run on the random field ising model defined on random regular graphs in the zero temperature limit we introduce the notion of extremal solutions for the bp equations and we use them to fix a fraction of spins in their ground state configuration at the phase transition point the fraction of unconstrained spins percolates and their number diverges with the system size this in turn makes the associated optimization problem highly non trivial in the critical region using the bounds on the bp messages provided by the extremal solutions we design a new and very easy to implement bp scheme which is able to output a large number of stable fixed points on one hand this new algorithm is able to provide the minimum energy configuration with high probability in a competitive time on the other hand we found that the number of fixed points of the bp algorithm grows with the system size in the critical region this unexpected feature poses new relevant questions about the physics of this class of models

statistical mechanics of maximal independent sets the graph theoretic concept of maximal independent set arises in several practical problems in computer science as well as in game theory a maximal independent set is defined by the set of occupied nodes that satisfy some packing and covering constraints it is known that finding minimum and maximum density maximal independent sets are hard optimization problems in this paper we use cavity method of statistical physics and monte carlo simulations to study the corresponding constraint satisfaction problem on random graphs we obtain the entropy of maximal independent sets within the replica symmetric and one step replica symmetry breaking frameworks shedding light on the metric structure of the landscape of solutions and suggesting a class of possible algorithms this is of particular relevance for the application to the study of strategic interactions in social and economic networks where maximal independent sets correspond to pure nash equilibria of a graphical game of public goods allocation

a better algorithm for random k sat let boldsymbol phi be a uniformly distributed random k sat formula with n variables and m clauses we present a polynomial time algorithm that finds a satisfying assignment of boldsymbol phi with high probability for constraint densities m n 1 varepsilon k 2 k ln k k where varepsilon k rightarrow0 previously no efficient algorithm was known to find satisfying assignments with a nonvanishing probability beyond m n 1 817 cdot2 k k a frieze and s suen j algorithms 20 1996 pp 312 355

fluctuations in the random link matching problem using the replica approach and the cavity method we study the fluctuations of the optimal cost in the random link matching problem by means of replica arguments we derive the exact expression of its variance moreover we study the large deviation function deriving its expression in two different ways namely using both the replica method and the cavity method

constructing free energy approximations and generalized belief propagation algorithms important inference problems in statistical physics computer vision error correcting coding theory and artificial intelligence can all be reformulated as the computation of marginal probabilities on factor graphs the belief propagation bp algorithm is an efficient way to solve these problems that is exact when the factor graph is a tree but only approximate when the factor graph has cycles we show that bp fixed points correspond to the stationary points of the bethe approximation of the free energy for a factor graph we explain how to obtain region based free energy approximations that improve the bethe approximation and corresponding generalized belief propagation gbp algorithms we emphasize the conditions a free energy approximation must satisfy in order to be a valid or maxent normal approximation we describe the relationship between four different methods that can be used to generate valid approximations the bethe method the junction graph method the cluster variation method and the region graph method finally we explain how to tell whether a region based approximation and its corresponding gbp algorithm is likely to be accurate and describe empirical results showing that gbp can significantly outperform bp

random subcubes as a toy model for constraint satisfaction problems we present an exactly solvable random subcube model inspired by the structure of hard constraint satisfaction and optimization problems our model reproduces the structure of the solution space of the random k satisfiability and k coloring problems and undergoes the same phase transitions as these problems the comparison becomes quantitative in the large k limit distance properties as well the x satisfiability threshold are studied the model is also generalized to define a continuous energy landscape useful for studying several aspects of glassy dynamics

typology of phase transitions in bayesian inference problems many inference problems undergo phase transitions as a function of the signal to noise ratio a prominent example of this phenomenon being found in the stochastic block model sbm that generates a random graph with a hidden community structure some of these phase transitions affect the information theoretic optimal but possibly computationally expensive estimation procedure others concern the behavior of efficient iterative algorithms a computational gap opens when the phase transitions for these two aspects do not coincide leading to a hard phase in which optimal inference is computationally challenging in this paper we refine this description in two ways from a qualitative perspective we emphasize the existence of more generic phase diagrams with a hybrid hard phase in which it is computationally easy to reach a nontrivial inference accuracy but computationally hard to match the information theoretic optimal one we support this discussion by quantitative expansions of the functional cavity equations that describe inference problems on sparse graphs around their trivial solution these expansions shed light on the existence of hybrid hard phases for a large class of planted constraint satisfaction problems and on the question of the tightness of the kesten stigum ks bound for the associated tree reconstruction problem our results show that the instability of the trivial fixed point is not generic evidence for the bayes optimality of the message passing algorithms we clarify in particular the status of the symmetric sbm with four communities and of the tree reconstruction of the associated potts model in the assortative ferromagnetic case the ks bound is always tight whereas in the disassortative antiferromagnetic case we exhibit an explicit criterion involving the degree distribution that separates a large degree regime where the ks bound is tight and a low degree regime where it is not we also investigate the sbm with two communities of different sizes also known as the asymmetric ising model and describe quantitatively its computational gap as a function of its asymmetry as well as a version of the sbm with two groups of q 1 and q 2 communities we complement this study with numerical simulations of the belief propagation iterative algorithm confirming that its behavior on large samples is well described by the cavity method

on two invariants of three manifolds from hopf algebras abstract we prove a 20 year old conjecture concerning two quantum invariants of three manifolds that are constructed from finite dimensional hopf algebras namely the kuperberg invariant and the hennings kauffman radford invariant the two invariants can be viewed as a non semisimple generalization of the turaev viro barrett westbury tvbw invariant and the witten reshetikhin turaev wrt invariant respectively by a classical result relating tvbw and wrt it follows that the kuperberg invariant for a semisimple hopf algebra is equal to the hennings kauffman radford invariant for the drinfeld double of the hopf algebra however whether the relation holds for non semisimple hopf algebras has remained open partly because the introduction of framings in this case makes the kuperberg invariant significantly more complicated to handle we give an affirmative answer to this question an important ingredient in the proof involves using a special heegaard diagram in which one family of circles gives the surgery link of the three manifold represented by the heegaard diagram

on the connectivity of cobordisms and half projective tqft s we consider a generalization of the axioms of a tqft the so called half projective tqft s where we inserted an anomaly in the composition law here 0 is a coboundary in a group cohomological sense on the cobordism categories with non negative integer values the element of the ring over which the tqft is defined does not have to be invertible in particular it may be zero

a unified witten reshetikhin turaev invariant for integral homology spheres we construct an invariant j m of integral homology spheres m with values in a completion widehat mathbb z q of the polynomial ring q such that the evaluation at each root of unity gives the the su 2 witten reshetikhin turaev invariant m of m at thus j m unifies all the su 2 witten reshetikhin turaev invariants of m it also follows that m as a function on behaves like an analytic function defined on the set of roots of unity

modular group representations and fusion in logarithmic conformal field theories and in the quantum group center the sl 2 representation on the center of the restricted quantum group open image in new window at the primitive 2pth root of unity is shown to be equivalent to the sl 2 representation on the extended characters of the logarithmic 1 p conformal field theory model the multiplicative jordan decomposition of the open image in new window ribbon element determines the decomposition of into a pointwise product of two commuting sl 2 representations one of which restricts to the grothendieck ring this restriction is equivalent to the sl 2 representation on the 1 p characters related to the fusion algebra via a nonsemisimple verlinde formula the grothendieck ring of open image in new window at the primitive 2pth root of unity is shown to coincide with the fusion algebra of the 1 p logarithmic conformal field theory model as a by product we derive q binomial identities implied by the fusion algebra realized in the center of open image in new window

gibbs states and the set of solutions of random constraint satisfaction problems abstract an instance of a random constraint satisfaction problem defines a random subset the set of solutions of a large product space x n the set of assignments we consider two prototypical problem ensembles random k satisfiability and q coloring of random regular graphs and study the uniform measure with support on s as the number of constraints per variable increases this measure first decomposes into an exponential number of pure states clusters and subsequently condensates over the largest such states above the condensation point the mass carried by the n largest states follows a poisson dirichlet process for typical large instances the two transitions are sharp we determine their precise location further we provide a formal definition of each phase transition in terms of different notions of correlation between distinct variables in the problem the degree of correlation naturally affects the performances of many search sampling algorithms empirical evidence suggests that local monte carlo markov chain strategies are effective up to the clustering phase transition and belief propagation up to the condensation point finally refined message passing techniques such as survey propagation may also beat this threshold message passing algorithms phase transitions random graphs

clustering of solutions in the random satisfiability problem using elementary rigorous methods we prove the existence of a clustered phase in the random k sat problem for k geq 8 in this phase the solutions are grouped into clusters which are far away from each other the results are in agreement with previous predictions of the cavity method and give a rigorous confirmation to one of its main building blocks it can be generalized to other systems of both physical and computational interest

region graph partition function expansion and approximate free energy landscapes theory and some numerical results graphical models for finite dimensional spin glasses and real world combinatorial optimization and satisfaction problems usually have an abundant number of short loops the cluster variation method and its extension the region graph method are theoretical approaches for treating the complicated short loop induced local correlations for graphical models represented by non redundant or redundant region graphs approximate free energy landscapes are constructed in this paper through the mathematical framework of region graph partition function expansion several free energy functionals are obtained each of which use a set of probability distribution functions or functionals as order parameters these probability distribution function functionals are required to satisfy the region graph belief propagation equation or the region graph survey propagation equation to ensure vanishing correction contributions of region subgraphs with dangling edges as a simple application of the general theory we perform region graph belief propagation simulations on the square lattice ferromagnetic ising model and the edwards anderson model considerable improvements over the conventional bethe peierls approximation are achieved collective domains of different sizes in the disordered and frustrated square lattice are identified by the message passing procedure such collective domains and the frustrations among them are responsible for the low temperature glass like dynamical behaviors of the system

threshold values stability analysis and high q asymptotics for the coloring problem on random graphs we consider the problem of coloring erdos renyi and regular random graphs of finite connectivity using q colors it has been studied so far using the cavity approach within the so called one step replica symmetry breaking 1rsb ansatz we derive a general criterion for the validity of this ansatz and applying it to the ground state we provide evidence that the 1rsb solution gives exact threshold values c q for the transition from the colorable to the uncolorable phase with q colors we also study the asymptotic thresholds for q 1 finding c q 2q ln q ln q 1 o 1 in perfect agreement with rigorous mathematical bounds as well as the nature of excited states and give a global phase diagram of the problem

hiding quiet solutions in random constraint satisfaction problems we study constraint satisfaction problems on the so called planted random ensemble we show that for a certain class of problems e g graph coloring many of the properties of the usual random ensemble are quantitatively identical in the planted random ensemble we study the structural phase transitions and the easy hard easy pattern in the average computational complexity we also discuss the finite temperature phase diagram finding a close connection with the liquid glass solid phenomenology

minimizing unsatisfaction in colourful neighbourhoods colouring sparse graphs under various restrictions is a theoretical problem of significant practical relevance here we consider the problem of maximizing the number of different colours available at the nodes and their neighbourhoods given a predetermined number of colours in the analytical framework of a tree approximation carried out at both zero and finite temperatures solutions obtained by population dynamics give rise to estimates of the threshold connectivity for the incomplete to complete transition which are consistent with those of existing algorithms the nature of the transition as well as the validity of the tree approximation are investigated

three manifold invariants associated with restricted quantum groups we show a simple relation between witten reshetikhin turaev su 2 invariant and the hennings invariant associated with the restricted quantum mathfrak sl 2 these invariants are defined in very different methods the former uses the representation theory of quantum mathfrak sl 2 while the latter uses the integral of the hopf algebra but they turn out to be the same for most rational homology 3 spheres up to a sign this relation can be used to prove the integrality of the former invariant

sharp thresholds of graph properties and the sat problem consider g n p to be the probability space of random graphs on n vertices with edge probability p we will be considering subsets of this space defined by monotone graph properties a monotone graph property p is a property of graphs such that a p is invariant under graph automorphisims b if graph h has property p then so does any graph g having h as a subgraph a monotone symmetric family of graphs is a family defined by such a property one of the first observations made about random graphs by erdos and renyi in their seminal work on random graph theory 12 was the existence of threshold phenomena the fact that for many interesting properties p the probability of p appearing in g n p exhibits a sharp increase at a certain critical value of the parameter p bollobas and thomason proved the existence of threshold functions for all monotone set properties 6 and in 14 it is shown that this behavior is quite general and that all monotone graph properties exhibit threshold behavior i e the probability of their appearance increases from values very close to 0 to values close to 1 in a very small interval more precise analysis of the size of the threshold interval is done in 7 this threshold behavior which occurs in various settings which arise in combinatorics and computer science is an instance of the phenomenon of phase transitions which is the subject of much interest in statistical physics one of the main questions that arises in studying phase transitions is how sharp is the transition for example one of the motivations for this paper arose from the question of the sharpness of the phase transition for the property of satisfiability of a random kcnf boolean formula nati linial who introduced me to this problem suggested that although much concrete analysis was being performed on this problem the best approach would be to find general conditions for sharpness of the phase transition answering the question posed in 14 as to the relation between the length of the threshold interval and the value of the critical probability in this paper we indeed introduce a simple condition and prove it is sufficient stated roughly in the setting of random graphs the main theorem states that if a property has a coarse threshold then it can be approximated by the property of having certain given graphs as a subgraph this condition can be applied in a more

coloring random graphs we study the graph coloring problem over random graphs of finite average connectivity c given a number q of available colors we find that graphs with low connectivity admit almost always a proper coloring whereas graphs with high connectivity are uncolorable depending on q we find the precise value of the critical average connectivity c q moreover we show that below c q there exists a clustering phase c e c d c q in which ground states spontaneously divide into an exponential number of clusters and where the proliferation of metastable states is responsible for the onset of complexity in local search algorithms

the freezing threshold for k colourings of a random graph we determine the exact value of the freezing threshold rfk for k colourings of a random graph when k 14 we prove that for random graphs with density above rfk almost every colouring is such that a linear number of vertices are frozen meaning that their colour cannot be changed by a sequence of alterations whereby we change the colours of o n vertices at a time always obtaining another proper colouring when the density is below rfk then almost every colouring is such that every vertex can be changed by a sequence of alterations where we change o log n vertices at a time frozen vertices are a key part of the clustering phenomena discovered using methods from statistical physics the value of the freezing threshold was previously determined by the nonrigorous cavity method

a model with darwinian dynamics on a rugged landscape we discuss the population dynamics with selection and random diffusion keeping the total population constant in a fitness landscape associated with constraint satisfaction a paradigm for difficult optimization problems we obtain a phase diagram in terms of the size of the population and the diffusion rate with a glass phase inside which the dynamics keeps searching for better configurations and outside which deleterious mutations spoil the performance the phase diagram is analogous to that of dense active matter in terms of temperature and drive

stability analysis on the finite temperature replica symmetric and first step replica symmetry broken cavity solutions of the random vertex cover problem the vertex cover problem is a prototypical hard combinatorial optimization problem it was studied in recent years by physicists using the cavity method of statistical mechanics in this paper the stability of the finite temperature replica symmetric rs and the first step replica symmetry broken 1rsb cavity solutions of the vertex cover problem on random regular graphs of finite vertex degree k are analyzed by population dynamics simulations we found that 1 the lowest temperature for the rs solution to be stable t rs k is not a monotonic function of k 2 at relatively large connectivity k and temperature t slightly below the dynamic transition temperature t d k the 1rsb solutions with small but non negative complexity values are stable and 3 the dynamical transition temperature t d and kauzmann temperature t k is equal to each other similar results are obtained on random poissonian graphs

the asymptotic k sat threshold abstract since the early 2000s physicists have developed an ingenious but non rigorous formalism called the cavity method to put forward precise conjectures on phase transitions in random problems mezard et al 2002 37 the cavity method predicts that the satisfiability threshold in the random k sat problem is r k sat 2 k ln 2 1 2 1 ln 2 e k with lim k e k 0 mertens et al 2006 35 this paper contains a proof of the conjecture