colored group field theory random matrix models generalize to group field theories gft whose feynman graphs are dual to higher dimensional topological spaces the perturbative development of the usual gft s is rather involved combinatorially and plagued by topological singularities which we discuss in great detail in this paper thus very difficult to control and unsatisfactory

average path length in random networks analytic solution for the average path length in a large class of uncorrelated random networks with hidden variables is found we apply the approach to classical random graphs of erdos and renyi er evolving networks introduced by barabasi and albert as well as random networks with asymptotic scale free connectivity distributions characterized by an arbitrary scaling exponent alpha 2 our result for 2 infinity there is a saturation effect for the average path length

the 1 n expansion of tensor models beyond perturbation theory we analyze in full mathematical rigor the most general quartically perturbed invariant probability measure for a random tensor using a version of the loop vertex expansion which we call the mixed expansion we show that the cumulants write as explicit series in 1 n plus bounded rest terms the mixed expansion recasts the problem of determining the subleading corrections in 1 n into a simple combinatorial problem of counting trees decorated by a finite number of loop edges

asymptotics of discrete orthogonal polynomials and the continuum limit of the toda lattice in a recent monograph deift and mclaughlin considered the continuum limit of the toda lattice and solved the problem using perturbation techniques such as the wentzel kramers brillouin wkb method we present an alternative approach based on the analytic theory of discrete orthogonal polynomials the method is based on some extremal problems in the theory of logarithmic potentials and relies on some results for the asymptotic theory of orthogonal polynomials with a discrete orthogonality measure

the average distances in random graphs with given expected degrees abstract random graph theory is used to examine the small world phenomenon any two strangers are connected through a short chain of mutual acquaintances we will show that for certain families of random graphs with given expected degrees the average distance is almost surely of order log n log d where d is the weighted average of the sum of squares of the expected degrees of particular interest are power law random graphs in which the number of vertices of degree k is proportional to 1 k for some fixed exponent for the case of 3 we prove that the average distance of the power law graphs is almost surely of order log n log d however many internet social and citation networks are power law graphs with exponents in the range 2 3 for which the power law random graphs have average distance almost surely of order log log n but have diameter of order log n provided having some mild constraints for the average distance and maximum degree in particular these graphs contain a dense subgraph which we call the core having nc log log n vertices almost all vertices are within distance log log n of the core although there are vertices at distance log n from the core

renormalization theory in four dimensional scalar fields i we present a renormalization group approach to the renormalization theory of 44 using techniques that have been introduced and used in previous papers and that lead to very simple methods to bound the coefficients of the effective potential and of the schwinger functions the main aim of this paper is to show how one can in this way obtain then bounds

new integrable systems related to the relativistic toda lattice the bi hamiltonian structure of the relativistic toda lattice is exploited to introduce some new integrable lattice systems their integrable discretizations are obtained by means of the general procedure proposed recently by the author backlund transformations between the new systems and the relativistic toda lattice in both the continuous and discrete time formulations are established

a continuum limit of the relativistic toda lattice asymptotic theory of discrete laurent orthogonal polynomials with varying recurrence coefficients we consider the continuum limit of the relativistic toda lattice in particular we propose a method in order to integrate this system of nonlinear partial differential equations for some particular initial data and boundary conditions before possible shocks first we recall the relation between the finite relativistic toda lattice and the theory of discrete laurent orthogonal polynomials our analysis is then based on some results for the asymptotic theory of discrete laurent orthogonal polynomials with varying recurrence coefficients and the connection with a constrained and weighted extremal problem for logarithmic potentials

the semiclassical limit of the defocusing nls hierarchy we establish the semiclassical limit of the one dimensional defocusing cubic nonlinear schrodinger nls equation complete integrability is exploited to obtain a global characterization of the weak limits of the entire nls hierarchy of conserved densities as the field evolves from reflectionless initial data under all the associated commuting flows consequently this also establishes the zero dispersion limit of the modified korteweg de vries equation that resides in that hierarchy we have adapted and clarified the strategy introduced by lax and levermore to study the zero dispersion limit of the korteweg de vries equation expanding it to treat entire integrable hierarchies and strengthening the limits obtained a crucial role is played by the convexity of the underlying log determinant with respect to the times associated with the commuting flows c 1999 john wiley sons inc

orthogonal laurent polynomials and strong moment theory a survey these topics are found in many parts of 20th century mathematics and its applications in mathematical physics chemistry statistics and engineering historically the analytic theory of continued fractions has played a central role in both the origin and the development of the other topics continued fractions are intimately related to pad e approximants and special functions emphasis is given to the development of strong moment theory and orthogonal laurent polynomials and to the related continued fractions quadrature formulas integral transforms and linear functionals by a strong moment problem we mean the following for a given bisequence n n of real numbers does there exist a distribution function such that

average hopcount of the shortest path in tree like components with finite size abstract an exact formula for computing the average hopcount of the shortest path in finite size tree like components of undirected unweighted random networks is proposed in a tree like component with size s there exists virtually only one shortest path between two arbitrary nodes the summation of hopcounts of all shortest paths can be calculated approximately by the summation of the betweenness of all nodes and the difference between them is only a constant s s 1 therefore the average hopcount can be calculated by further dividing the summation by the number of all shortest paths in this paper we first derive the conditional probability p k s of the degree distribution of finite components with size s and the summation of all nodal betweenness respectively by means of these results we obtain the exact formula for calculating the average hopcount also we confirm the proposed formula by simulations for networks with poisson and power law degree distributions respectively

a polynomial of graphs on surfaces i e graphs realized as disks vertices joined together by strips edges glued to their boundaries corresponding to neighbourhoods of graphs embedded into surfaces we construct a four variable polynomial invariant of these objects the ribbon graph polynomial which has all the main properties of the tutte polynomial although the ribbon graph polynomial extends the tutte polynomial its definition is very different and it depends on the topological structure in an essential way

attack vulnerability of complex networks we study the response of complex networks subject to attacks on vertices and edges several existing complex network models as well as real world networks of scientific collaborations and internet traffic are numerically investigated and the network performance is quantitatively measured by the average inverse geodesic length and the size of the largest connected subgraph for each case of attacks on vertices and edges four different attacking strategies are used removals by the descending order of the degree and the betweenness centrality calculated for either the initial network or the current network during the removal procedure it is found that the removals by the recalculated degrees and betweenness centralities are often more harmful than the attack strategies based on the initial network suggesting that the network structure changes as important vertices or edges are removed furthermore the correlation between the betweenness centrality and the degree in complex networks is studied

renormalizable models in rank d geq 2 tensorial group field theory classes of renormalizable models in the tensorial group field theory framework are investigated the rank d tensor fields are defined over d copies of a group manifold g d u 1 d or g d su 2 d with no symmetry and no gauge invariance assumed on the fields in particular we explore the space of renormalizable models endowed with a kinetic term corresponding to a sum of momenta of the form p 2a a in 0 1 this study is tailored for models equipped with laplacian dynamics on gd case a 1 but also for more exotic nonlocal models in quantum topology case 0 a 1 a generic model can be written dim g d phi k d a where k is the maximal valence of its interactions using a multi scale analysis for the generic situation we identify several classes of renormalizable actions including matrix model actions in this specific instance we find a tower of renormalizable matrix models parametrized by k geq 4 in a second part of this work we study the uv behavior of the models up to maximal valence of interaction k 6 all rank d geq 3 tensor models proved renormalizable are asymptotically free in the uv all matrix models with k 4 have a vanishing function at one loop and very likely reproduce the same feature of the grosse wulkenhaar model commun math phys 256 305 2005

the small dispersion limit of the korteweg de vries equation ii in part i the scattering transform method is used to study the weak limit of solutions to the initial value problem for the korteweg devries kdv equation as the dispersion tends to zero in that limit the associated schrodinger operator becomes semiclassical so the exact scattering data is replaced by its corresponding wkb expressions only nonpositive initial data are considered in that case the limiting reflection coefficient vanishes the explicit solution of kay and moses for the reflectionless inverse transform is then analyzed and the weak limit valid for all time is characterized by a quadratic minimum problem with constraints this minimum problem is reduced to a riemann hilbert problem in function theory in parts ii and iii we use function theoretical methods to solve the riemann hilbert problem in terms of solutions to an auxiliary initial value problem the weak limit satisfies the kdv equation with the dispersive term dropped until its derivatives become infinite up to that time the weak limit is a strong l2 limit at later times the weak limit is locally described by whitham s averaged equations or more generally by the equations found by flaschka et al using multiphase averaging for large times behavior of the weak limit is studied directly from the minimum problem

superlinear convergence of the rational arnoldi method for the approximation of matrix functions a superlinear convergence bound for rational arnoldi approximations to functions of matrices is derived this bound generalizes the well known superlinear convergence bound for the conjugate gradient method to more general functions with finite singularities and to rational krylov spaces a constrained equilibrium problem from potential theory is used to characterize a max min quotient of a nodal rational function underlying the rational arnoldi approximation where an additional external field is required for taking into account the poles of the rational krylov space the resulting convergence bound is illustrated at several numerical examples in particular the convergence of the extended krylov method for the matrix square root

betweenness centrality of an edge in tree like components with finite size using the generating function formalism we propose an exact formula for estimating betweenness centrality of an edge in finite tree like components of random networks with arbitrary degree distributions as a function of the degrees of two vertices interconnected by the edge and the size of the component to which the edge belongs the formula can give an estimation of the mean value of the edge betweenness we confirm the formula by simulations for poisson and power law distributions respectively we also study the asymptotic behavior of the formula and find that there is an asymptotic power law relation with an exponent 1 5 between the edge betweenness and the size of the component to which the edge belongs

two point pade expansions for a family of analytic functions abstract each member g z of a family of analytic functions defined by stieltjes transforms is shown to be represented by a positive t fraction the approximants of which form the main diagonal in the two point pade table of g z the positive t fraction is shown to converge to g z throughout a domain d a b z z b a uniformly on compact subsets in addition truncation error bounds are given for the approximants of the continued function these bounds supplement previously known bounds and apply in part of the domain of g z not covered by other bounds the proofs of our results employ properties of orthogonal l polynomials laurent polynomials and l gaussian quadrature which are of some interest in themselves a number of examples are considered

extremal polynomials on discrete sets we study asymptotics for orthogonal polynomials and other extremal polynomials on infinite discrete sets typical examples being the meixner polynomials and the charlier polynomials following ideas of rakhmanov dragnev and saff weshow that the asymptotic behaviour is governed by a constrained extremal energy problem for logarithmic potentials which can be solved explicitly we give formulas for the contracted zero distributions the n th root asymptotics and the asymptotics of the largest zeros

intersection theory on the moduli space of curves and the matrix airy function we show that two natural approaches to quantum gravity coincide this identity is nontrivial and relies on the equivalence of each approach to kdv equations we also investigate related mathematical problems

long time behavior of the continuum limit of the toda lattice and the generation of infinitely many gaps from c initial data we analyze a continuum limit of the finite non periodic toda lattice through an associated constrained maximization problem over spectral density functions the maximization problem was derived by deift and mclaughlin using the lax levermore approach initially developed for the zero dispersion limit of the kdv equation it encodes the evolution of the continuum limit for all times including evolution through shocks the formation of gaps in the support of the maximizer is indicative of oscillations in the toda lattice and the lack of strong convergence of the continuum limit for large times the maximizer tends to have zero gaps which is the continuum analogue of the sorting property of the finite lattice using methods from logarithmic potential theory we show that this behavior depends crucially on the initial data we exhibit initial data for which the zero gap ansatz holds uniformly in the spatial parameter at large times and other initial data for which this uniformity fails at all times we then construct an example of c smooth initial data generating at a later time infinitely many gaps in the support of the maximizer while for even larger times all gaps have closed

a discrete time relativistic toda lattice for each of the two simplest hamiltonian flows from the relativistic toda hierarchy we introduce two integrable symplectic discretizations all four discrete time systems are demonstrated to belong to the same hierarchy and to exemplify the general scheme for symplectic maps on groups equipped with quadratic poisson brackets the initial value problem for the difference equations is solved in terms of a factorization problem in a group interpolating hamiltonian flows are found for all maps

scaling limits of random planar maps with large faces we discuss asymptotics for large random planar maps under the assumption that the distribution of the degree of a typical face is in the domain of attraction of a stable distribution with index alpha in 1 2 when the number n of vertices of the map tends to infinity the asymptotic behavior of distances from a distinguished vertex is described by a random process called the continuous distance process which can be constructed from a centered stable process with no negative jumps and index alpha in particular the profile of distances in the map rescaled by the factor n 1 2 alpha converges to a random measure defined in terms of the distance process with the same rescaling of distances the vertex set viewed as a metric space converges in distribution as n to infty at least along suitable subsequences towards a limiting random compact metric space whose hausdorff dimension is equal to 2 alpha

orthogonal laurent polynomials and the strong hamburger moment problem abstract this paper is concerned with the strong hamburger moment problem shmp for a given double sequence of real numbers c c n does there exist a real valued bounded non decreasing function on with infinitely many points of increase such that for every integer n c n t n d t necessary and sufficient conditions for the existence of such a function are given in terms of the positivity of certain hankel determinants associated with c our approach is made through the study of orthogonal and quasi orthogonal laurent polynomials referred to here as l polynomials and closely related gaussian type quadrature formulas in the proof of sufficiency an inner product for l polynomials is defined in terms of the given double sequence c since orthogonal l polynomials are believed to be of interest in themselves some examples of specific systems are considered

the brownian map is the scaling limit of uniform random plane quadrangulations we prove that uniform random quadrangulations of the sphere with n faces endowed with the usual graph distance and renormalized by n 1 4 converge as n in distribution for the gromov hausdorff topology to a limiting metric space we validate a conjecture by le gall by showing that the limit is up to a scale constant the so called brownian map which was introduced by marckert mokkadem and le gall as the most natural candidate for the scaling limit of many models of random plane maps the proof relies strongly on the concept of geodesic stars in the map which are configurations made of several geodesics that only share a common endpoint and do not meet elsewhere

component sizes in networks with arbitrary degree distributions we give an exact solution for the complete distribution of component sizes in random networks with arbitrary degree distributions the solution tells us the probability that a randomly chosen node belongs to a component of size s for any s we apply our results to networks with the three most commonly studied degree distributions poisson exponential and power law as well as to the calculation of cluster sizes for bond percolation on networks which correspond to the sizes of outbreaks of epidemic processes on the same networks for the particular case of the power law degree distribution we show that the component size distribution itself follows a power law everywhere below the phase transition at which a giant component forms but takes an exponential form when a giant component is present

random graphs with arbitrary degree distributions and their applications recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the poisson degree distributions that have been widely studied in the past in this paper we develop in detail the theory of random graphs with arbitrary degree distributions in addition to simple undirected unipartite graphs we examine the properties of directed and bipartite graphs among other results we derive exact expressions for the position of the phase transition at which a giant component first forms the mean component size the size of the giant component if there is one the mean number of vertices a certain distance away from a randomly chosen vertex and the average vertex vertex distance within a graph we apply our theory to some real world graphs including the worldwide web and collaboration graphs of scientists and fortune 1000 company directors we demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world while in others there is a measurable discrepancy between theory and reality perhaps indicating the presence of additional social structure in the network that is not captured by the random graph

class of correlated random networks with hidden variables we study a class of models of correlated random networks in which vertices are characterized by hidden variables controlling the establishment of edges between pairs of vertices we find analytical expressions for the main topological properties of these models as a function of the distribution of hidden variables and the probability of connecting vertices the expressions obtained are checked by means of numerical simulations in a particular example the general model is extended to describe a practical algorithm to generate random networks with an a priori specified correlation structure we also present an extension of the class to map nonequilibrium growing networks to networks with hidden variables that represent the time at which each vertex was introduced in the system

betweenness centrality in finite components of complex networks abstract we use generating function formalism to obtain an exact formula of the betweenness centrality in finite components of random networks with arbitrary degree distributions the formula is obtained as a function of the degree and the component size and is confirmed by simulations for poisson exponential and power law degree distributions we find that the betweenness centralities for the three distributions are asymptotically power laws with an exponent 1 5 and are invariant to the particular distribution parameters

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

universality for the distance in finite variance random graphs we generalize the asymptotic behavior of the graph distance between two uniformly chosen nodes in the configuration model to a wide class of random graphs among others this class contains the poissonian random graph the expected degree random graph and the generalized random graph including the classical erdos renyi graph in the paper we assign to each node a deterministic capacity and the probability that there exists an edge between a pair of nodes is equal to a function of the product of the capacities of the pair divided by the total capacity of all the nodes we consider capacities which are such that the degrees of a node have uniformly bounded moments of order strictly larger than two so that in particular the degrees have finite variance we prove that the graph distance grows like log nu n where the nu depends on the capacities and n denotes the size of the graph in addition the random fluctuations around this asymptotic mean log nu n are shown to be tight we also consider the case where the capacities are independent copies of a positive random lambda with p lambda x 3 againg resulting in graphs where the degrees have finite variance the method of proof of these results is to couple each member of the class to the poissonian random graph for which we then give the complete proof by adapting the arguments of van der hofstad et al

renormalization of tensorial group field theories abelian u 1 models in four dimensions we tackle the issue of renormalizability for tensorial group field theories tgft including gauge invariance conditions with the rigorous tool of multi scale analysis to prepare the ground for applications to quantum gravity models in the process we define the appropriate generalization of some key qft notions including connectedness locality and contraction of high subgraphs we also define a new notion of wick ordering corresponding to the subtraction of maximal melonic tadpoles we then consider the simplest examples of dynamical 4 dimensional tgft with gauge invariance conditions for the abelian u 1 case we prove that they are super renormalizable for any polynomial interaction