end to end distance from the green s function for a hierarchical self avoiding walk in four dimensions in bei92 we introduced a levy process on a hierarchical lattice which is four dimensional in the sense that the green s function for the process equals frac 1 x 2 if the process is modified so as to be weakly self repelling it was shown that at the critical killing rate mass squared c the green s function behaves like the free one now we analyze the end to end distance of the model and show that its expected value grows as a constant times sqrt t log frac 1 8 t left 1 o left frac log log t log t right right which is the same law as has been conjectured for self avoiding walks on the simple cubic lattice 4 the proof uses inverse laplace transforms to obtain the end to end distance from the green s function and requires detailed properties of the green s function throughout a sector of the complex plane these estimates are derived in a companion paper bi02

brief minimal and non minimal optimal fixed order compensators for time varying discrete time systems using the minimality property of finite horizon time varying compensators established in this paper and the moore penrose pseudo inverse instead of the standard inverse strengthened discrete time optimal projection equations sdope and associated boundary conditions are derived for finite horizon fixed order lqg compensation they constitute a two point boundary value problem explicit in the lqg problem parameters which is equivalent to first order necessary optimality conditions and which is suitable for numerical solution the minimality property implies that minimal compensators have time varying dimensions and that the finite horizon optimal full order compensator is not minimal the use of the moore penrose pseudo inverse is further exploited to reveal that the optimal projection approach can be generalised but only to partially include non minimal compensators furthermore the structure of the space of optimal compensators with arbitrary dimensions is revealed to a large extent max min compensator dimensions are introduced and their significance in solving numerically the two point boundary value problem is explained the numerical solution is presented in a recently published companion paper which relies on the results of this paper

large momentum properties and wilson short distance expansion in non perturbative field theory large momentum properties and wilson zimmerman short distance expansion are established via phase space analysis for the weakly coupled massive gross neveu model in dimension 2 methods are applicable more generally

random growth urban model with geographical fitness abstract this paper formulates a random growth urban model with a notion of geographical fitness using techniques of complex network theory we study our system as a type of preferential attachment model with fitness and we analyze its macro behavior to clarify the properties of the city size distributions it predicts first restricting the geographical fitness to take positive values and using a continuum approach we show that the city size distributions predicted by our model asymptotically approach pareto distributions with coefficients greater than unity then allowing the geographical fitness to take negative values we perform local coefficient analysis to show that the predicted city size distributions can deviate from pareto distributions as is often observed in actual city size distributions as a result the model we propose can generate a generic class of city size distributions including but not limited to pareto distributions for applications to city population projections our simple model requires randomness only when new cities are created not during their subsequent growth this property leads to smooth trajectories of city population growth in contrast to other models using gibrat s law in addition a discrete form of our dynamical equations can be used to estimate past city populations based on present day data this fact allows quantitative assessment of the performance of our model further study is needed to determine appropriate formulas for the geographical fitness

scaling properties of scale free evolving networks continuous approach the scaling behavior of scale free evolving networks arising in areas such as communications scientific citations collaborations etc is studied we derive universal scaling relations describing properties of such networks and indicate the limits of their validity we show that the main properties of scale free evolving networks may be described in the framework of a simple continuous approach the simplest models of networks growing according to a mechanism of preferential attachment of links to nodes are used we consider different forms of this preference and demonstrate that the range of preferential attachments producing scale free networks is wide we also obtain scaling relations for networks with nonlinear accelerating growth and describe the temporal evolution of the arising distributions size effects char22 the cutoffs of these distributions char22 introduce restrictions for the observation of power law dependences mainly we discuss the so called degree distribution i e the distribution of the number of connections of nodes a scaling form of the distribution of links between pairs of individual nodes for a growing network of citations is also studied we describe the effects of differences between nodes the aging of nodes changes the exponents of the distributions the appearance of a single node with high fitness changes the degree distribution of a network dramatically if its fitness exceeds some threshold value this node captures a finite part of all links of the network we show that permanent random damage to a growing scale free network char22 a permanent deletion of some links char22 radically changes the values of the scaling exponents results of other kinds of permanent damage are described

gaussian behavior of loop erased self avoiding random walk in four dimensions une marche auto evitante de longueur n sur le reseau entier z d est une suite ordonnee de points x 0 0 x 1 x n avec x i x i 1 1 et x i x j pour i j on etudie le comportement d une telle marche a d 4 quand on applique le processus de boucle effacee

on the fock space representation of functionals of the occupation field and their renormalization on fournit une representation du champ d occupation et de ses puissances renormalisees dans un bon espace de fock

the broken supersymmetry phase of a self avoiding random walk we consider a weakly self avoiding random walk on a hierarchical lattice ind 4 dimensions we show that for choices of the killing ratea less than the critical valueacthe dominant walks fill space which corresponds to a spontaneously broken supersymmetry phase we identify the asymptotic density to which walks fill space a to be a supersymmetric order parameter for this transition we prove that a ac a log ac a 1 2 asa ac which is mean field behavior with logarithmic corrections as expected for a system in its upper critical dimension

multiple power law structures in heterogeneous complex networks this paper develops a framework for analyzing and designing dynamic networks comprising different classes of nodes that coexist and interact in one shared environment we consider ad hoc i e nodes can leave the network unannounced and no node has any global knowledge about the class identities of other nodes preferentially grown networks where different classes of nodes are characterized by different sets of local parameters used in the stochastic dynamics that all nodes in the network execute we show that multiple scale free structures one within each class of nodes and with tunable power law exponents as determined by the sets of parameters characterizing each class emerge naturally in our model moreover the coexistence of the scale free structures of the different classes of nodes can be captured by succinct phase diagrams which show a rich set of structures including stable regions where different classes coexist in heavy tailed i e the exponent is between 2 and 3 and light tailed i e the exponent is greater than 3 states and sharp phase transitions the topology of the emergent networks is also shown to display a complex structure akin to the distribution of different components of an alloyed material e g nodes with a light tailed scale free structure get embedded to the outside of the network and have most of their edges connected to nodes belonging to the class with a heavy tailed distribution finally we show how the dynamics formulated in this paper will serve as an essential part of ad hoc networking protocols which can lead to the formation of robust and efficiently searchable networks including the well known peer to peer networks even under very dynamic conditions

survival of contact processes on the hierarchical group we consider contact processes on the hierarchical group where sites infect other sites at a rate depending on their hierarchical distance and sites become healthy with a constant recovery rate if the infection rates decay too fast as a function of the hierarchical distance then we show that the critical recovery rate is zero on the other hand we derive sufficient conditions on the speed of decay of the infection rates for the process to exhibit a nontrivial phase transition between extinction and survival for our sufficient conditions we use a coupling argument that compares contact processes on the hierarchical group with freedom two with contact processes on a renormalized lattice an interesting novelty in this renormalization argument is the use of a result due to rogers and pitman on markov functionals

connectivity degrees in the threshold preferential attachment model abstract in this paper we present a study of the connectivity degrees of the threshold preferential attachment model a generalization of the barabasi albert model to heterogeneous complex networks the threshold model incorporates the states of the nodes in its preferential linking rule and assumes that the affinity between network nodes follows an inverse relationship with the distance between their states we numerically analyze the connectivity degrees of the model studying the influence of the main parameters on the distribution of connectivity degrees and its statistics the average degree and highest degree of the network we show that such statistics exhibit markedly different behaviors in the dependence on the model parameters particularly as regards the interaction threshold nevertheless we show that the two statistics converge in the limit of null threshold and often exhibit scaling that can be described by power laws of the model parameters

the diffusive phase of a model of self interacting walks we consider simple random walk onzd perturbed by a factor exp t pjt wheret is the length of the walk and j t sum nolimits 0 leqslant i 0 ford 2 the diffusion constant is equal to 1 but ford 2 it is renormalized ford 1 andp 3 2 we prove diffusion for all real positive or negative ford 2 the scaling limit is brownian motion but ford 2 it is the edwards model with the wrong sign of the coupling when 0 which governs the limiting behaviour the latter arises since for p frac 4 d 2 t p j t is the discrete self intersection local time this establishes existence of a diffusive phase for this model existence of a collapsed phase for a very closely related model has been proven in work of bolthausen and schmock

green s function for a hierarchical self avoiding walk in four dimensions this is the second of two papers on the end to end distance of a weakly self repelling walk on a four dimensional hierarchical lattice it completes the proof that the expected value grows as a constant times sqrt t log frac 1 8 t left 1 o left frac log log t log t right right which is the same law as has been conjectured for self avoiding walks on the simple cubic lattice 4 apart from completing the program in the first paper the main result is that the green s function is almost equal to the green s function for the markov process with no self repulsion but at a different value of the killing rate which can be accurately calculated when the interaction is small furthermore the green s function is analytic in in a sector in the complex plane with opening angle greater than

a renormalisation group method i gaussian integration and normed algebras this paper is the first in a series devoted to the development of a rigorous renormalisation group method for lattice field theories involving boson fields fermion fields or both our immediate motivation is a specific model involving both boson and fermion fields which arises as a representation of the continuous time weakly self avoiding walk in this paper we define normed algebras suitable for a renormalisation group analysis and develop methods for performing analysis on these algebras we also develop the theory of gaussian integration on these normed algebras and prove estimates for gaussian integrals the concepts and results developed here provide a foundation for the continuation of the method presented in subsequent papers in the series

self avoiding walk in 5 or more dimensions using an expansion based on the renormalization group philosophy we prove that for a t step weakly self avoiding random walk in five or more dimensions the variance of the endpoint is of order t and the scaling limit is gaussian as t

markov processes as a tool in field theory abstract in a recent paper brydges frohlich and spencer have successfully applied markov chains to classical spin systems using probabilistic methods in a more systematic way we improve some of their results the methods can be extended to fields with continuous parameter such as p fields

dynamics of directed graphs the world wide web we introduce and simulate a growth model of the world wide web based on the dynamics of outgoing links that is motivated by the conduct of the agents in the real web to update outgoing links re directing them towards constantly changing selected nodes emergent statistical correlation between the distributions of outgoing and incoming links is a key feature of the dynamics of the web the growth phase is characterized by temporal fractal structures which are manifested in the hierarchical organization of links we obtain quantitative agreement with the recent empirical data in the real web for the distributions of in and out links and for the size of the connected component in a fully grown network of n nodes we study the structure of connected clusters of nodes that are accessible along outgoing links from a randomly selected node the distributions of size and depth of the connected clusters with a giant component exhibit supercritical behavior by decreasing the control parameter average fraction of updated and added links per time step towards c n 10 the web can resume a critical structure with no giant component in it we find a different universality class when the updates of links are not allowed i e for 0 corresponding to the network of science citations

reduced order lqg controllers for linear time varying plants abstract the problem of reduced order lqg optimization is addressed in a finite horizon linear time varying ltv system setting first order necessary conditions for local optimality in the parameter space is provided in terms of four coupled matrix differential equations this result provides a transparent generalization of the optimal projection equations of hyland and bernstein for the optimal steady state compensation of linear time invariant lti plants

on continuous time self avoiding random walk in dimension four we study the distribution of the end to end distance of continuous time self avoiding random walks ctrw in dimension four from two viewpoints from a real space renormalization group map on probabilities we conjecture the asymptotic behavior of the end to end distance of a weakly self avoiding random walk sarw that penalizes two body interactions of random walks in dimension four on a hierarchical lattice then we perform the monte carlo computer simulations of ctrw on the four dimensional integer lattice paying special attention to the difference in statistical behavior of the ctrw compared with the discrete time random walks in this framework we verify the result already predicted by the renormalization group method and provide new results related to enumeration of self avoiding random walks and calculation of the mean square end to end distance and gyration radius of continous time self avoiding random walks

the optimal projection equations for model reduction and the relationships among the methods of wilson skelton and moore first order necessary conditions for quadratically optimal reduced order modeling of linear time invariant systems are derived in the form of a pair of modified lyapunov equations coupled by an oblique projection which determines the optimal reduced order model this form of the necessary conditions considerably simplifies previous results of wilson 1 and clearly demonstrates the quadratic extremality and nonoptimality of the balancing method of moore 2 the possible existence of multiple solutions of the optimal projection equations is demonstrated and a relaxation type algorithm is proposed for computing these local extrema a component cost analysis of the model error criterion similar to the approach of skelton 3 is utilized at each iteration to direct the algorithm to the global minimum

the influence of preferential attachment on evolving networks abstract with the consideration of local events and preferential attachment the extended model proposed by albert and barabasi is studied firstly the extended model is built only by local events it is found that without preferential attachment the networks connectivity distribution decays exponentially for large connectivity and the decaying exponent varies with different probabilities of the addition of new links and rewiring secondly the extended model with partial preferential attachment is studied when preferential attachment only presents in the addition of new links or new nodes there is scale free regime in the phase diagram while preferential attachment is brought into the rewiring there coexist two regimes i e the exponential regime and the scale free regime thirdly fitness is introduced to the extended model with complete preferential attachment it is obtained that the network s connectivity distribution is a result affected by many factors including the fitness the local events and preferential attachment

a new discrete edwards model and a new polymer measure in two dimensions the new discrete edwards models in this paper are defined in terms of the so called restricted intersection local times of the lattice random walk in two dimensions we study the asymptotic behaviours of these new discrete edwards models in the superrenormalizable cases in particular by approximating these models we can construct new polymer measures in two dimensions which are different from the original polymer measures obtained by approximating the original discrete edwards models the new discrete edwards models can be thought of as zero component lattice 4 fields with different cutoffs in the free and interacting parts

a renormalisation group method v a single renormalisation group step this paper is the fifth in a series devoted to the development of a rigorous renormalisation group method applicable to lattice field theories containing boson and or fermion fields and comprises the core of the method in the renormalisation group method increasingly large scales are studied in a progressive manner with an interaction parametrised by a field polynomial which evolves with the scale under the renormalisation group map in our context the progressive analysis is performed via a finite range covariance decomposition perturbative calculations are used to track the flow of the coupling constants of the evolving polynomial but on their own perturbative calculations are insufficient to control error terms and to obtain mathematically rigorous results in this paper we define an additional non perturbative coordinate which together with the flow of coupling constants defines the complete evolution of the renormalisation group map we specify conditions under which the non perturbative coordinate is contractive under a single renormalisation group step our framework is essentially combinatorial but its implementation relies on analytic results developed earlier in the series of papers the results of this paper are applied elsewhere to analyse the critical behaviour of the 4 dimensional continuous time weakly self avoiding walk and of the 4 dimensional n component varphi 4 model in particular the existence of a logarithmic correction to mean field scaling for the susceptibility can be proved for both models together with other facts about critical exponents and critical behaviour

ballistic behavior in a 1d weakly self avoiding walk with decaying energy penalty we consider a weakly self avoiding walk in one dimension in which the penalty for visiting a site twice decays as exp t s p wheret ands are the times at which the common site is visited andp is a parameter we prove that ifp 3 2 then the walk should have diffusive behavior the proof and the heuristic argument make use of a real space renormalization group transformation

supersymmetry and the parisi sourlas dimensional reduction a rigorous proof functional integrals that are formally related to the average correlation functions of a classical field theory in the presence of random external sources are given a rigorous meaning their dimensional reduction to the schwinger functions of the corresponding quantum field theory in two fewer dimensions is proven this is done by reexpressing those functional integrals as expectations of a supersymmetric field theory the parisi sourlas dimensional reduction of a supersymmetric field theory to a usual quantum field theory in two fewer dimensions is proven

a non gaussian renormalization group fixed point for hierarchical scalar lattice field theories a rigorous method is developed to handle the large field problems in the wilson kadanoff renormalization group approach to critical lattice systems of unbounded spins we use this method to study in a hierarchical approximation the non gaussian renormalization group fixed point which governs the infrared behaviour of critical lattice field theories in three dimensions the method is an improvement of the analyticity techniques of gawedzki and kupiainen using borel summation techniques we are able to incorporate the large field region into the perturbative region so that the theory is completely described in terms of convergent expansions

the probability of intersection of independent random walks in four dimensions lets1 ands2 be independent simple random walks of lengthn inz4 starting at 0 andx0 respectively if x0 2 n it is shown that the probability that the paths intersect is of order logn 1 ifx0 0 it is shown that the probability of no intersection of the paths decays no faster than logn 1 and no slower than logn 1 2 it is conjectured that logn 1 2 is the actual decay rate

scale free networks with tunable degree distribution exponents we propose and study a model of scale free growing networks that gives a degree distribution dominated by a power law behavior with a model dependent hence tunable exponent the model represents a hybrid of the growing networks based on popularity driven and fitness driven preferential attachments as the network grows a newly added node establishes m new links to existing nodes with a probability p based on popularity of the existing nodes and a probability 1 p based on fitness of the existing nodes an explicit form of the degree distribution p p k is derived within a mean field approach for reasonably large k p p k approximately k gamma p f k p where the function f is dominated by the behavior of 1 ln k m for small values of p and becomes k independent as p 1 and gamma p is a model dependent exponent the degree distribution and the exponent gamma p are found to be in good agreement with results obtained by extensive numerical simulations

grad o perturbations of massless gaussian fields we investigate weak perturbations of the continuum massless gaussian measure by a class of approximately local analytic functionals and use our general results to give a new proof that the pressure of the dilute dipole gas is analytic in the activity

brief paper optimal reduced order compensation of time varying discrete time systems with deterministic and white parameters the finite horizon optimal compensation problem is considered in the case of linear time varying discrete time systems with deterministic and white stochastic parameters and quadratic criteria the dimensions of the compensator are a priori fixed and may be time varying also the dimensions of the system may be time varying strengthened discrete time optimal projection equations sdope are developed which within the class of minimal compensators are equivalent to the first order necessary optimality conditions based on the sdope and their associated boundary conditions two numerical algorithms are presented to solve the two point boundary value problem one is a homotopy algorithm while the second iterates the sdope repeatedly forward and backward in time the latter algorithm is much more efficient and constitutes a generalization of the single iteration of the control and estimation riccati equations associated with the full order problem for systems with deterministic parameters the algorithms are illustrated with a numerical example the case of systems with deterministic parameters will be treated as a special case of systems with white parameters

a simple method for finite range decomposition of quadratic forms and gaussian fields we present a simple method to decompose the green forms corresponding to a large class of interesting symmetric dirichlet forms into integrals over symmetric positive semi definite and finite range properly supported forms that are smoother than the original green form this result gives rise to multiscale decompositions of the associated gaussian free fields into sums of independent smoother gaussian fields with spatially localized correlations our method makes use of the finite propagation speed of the wave equation and chebyshev polynomials it improves several existing results and also gives simpler proofs

organization of growing random networks the organizational development of growing random networks is investigated these growing networks are built by adding nodes successively and linking each to an earlier node of degree k with an attachment probability a k when a k grows more slowly than linearly with k the number of nodes with k links n k t decays faster than a power law in k while for a k growing faster than linearly in k a single node emerges which connects to nearly all other nodes when a k is asymptotically linear n k t ensuremath sim mathrm tk ensuremath ensuremath nu with ensuremath nu dependent on details of the attachment probability but in the range 2l ensuremath nu l ensuremath infty the combined age and degree distribution of nodes shows that old nodes typically have a large degree there is also a significant correlation in the degrees of neighboring nodes so that nodes of similar degree are more likely to be connected the size distributions of the in and out components of the network with respect to a given node namely its descendants and ancestors are also determined the in component exhibits a robust s ensuremath 2 power law tail where s is the component size the out component has a typical size of order mathrm ln t and it provides basic insights into the genealogy of the network

the optimal projection equations for reduced order state estimation first order necessary conditions for optimal steady state reduced order state estimation for a linear time invariant plant in the presence of correlated disturbance and nonsingular measurement noise are derived in a new and highly simplified form in contrast to the lone matrix riccati equation arising in the full order kalman filter case the optimal steady state reduced order estimator is characterized by three matrix equations one modified riccati equation and two modified lyapunov equations coupled by a projection whose rank is precisely equal to the order of the estimator and which determines the optimal estimator gains this coupling is a graphic reminder of the suboptimality of proposed approaches involving either model reduction followed by full order estimator design or full order estimator design followed by estimator reduction techniques the results given here complement recently obtained results which characterize the optimal reduced order model by means of a pair of coupled modified lyapunov equations 7 and the optimal fixed order dynamic compensator by means of a coupled system of two modified riceati equations and two modified lyapunov equations 6

four dimensional weakly self avoiding walk with contact self attraction we consider the critical behaviour of the continuous time weakly self avoiding walk with contact self attraction on mathbb z 4 for sufficiently small attraction we prove that the susceptibility and correlation length of order p for any p 0 have logarithmic corrections to mean field scaling and that the critical two point function is asymptotic to a multiple of x 2 this shows that small contact self attraction results in the same critical behaviour as no contact self attraction a collapse transition is predicted for larger self attraction the proof uses a supersymmetric representation of the two point function and is based on a rigorous renormalisation group method that has been used to prove the same results for the weakly self avoiding walk without self attraction

human sexual contact network as a bipartite graph abstract a simple model to encapsulate the essential growth properties of the web of human sexual contacts is presented in the model only heterosexual connection is considered and represented by a random growing bipartite graph where both male female contact networks grow simultaneously the time evolution of the model is analysed by a rate equation approach leading to confirm that male and female sexual contact distributions decay as power laws with exponents depending on influx and charisma of the sexes

finite range decomposition for gaussian measures with improved regularity abstract we consider a family of gradient gaussian vector fields on the torus z l n z d adams koteck muller and independently bauerschmidt established the existence of a uniform finite range decomposition of the corresponding covariance operators i e the covariance can be written as a sum of covariance operators supported on increasing cubes with diameter l k we improve this result and show that the decay behaviour of the kernels in fourier space can be controlled then we show the regularity of the integration map that convolves functionals with the partial measures of the finite range decomposition in particular the new finite range decomposition avoids the loss of regularity which arises in the renormalisation group approach to anisotropic problems in statistical mechanics

monte carlo study of four dimensional self avoiding walks of up to one billion steps we study self avoiding walks on the four dimensional hypercubic lattice via monte carlo simulations of walks with up to one billion steps we study the expected logarithmic corrections to scaling and find convincing evidence in support the scaling form predicted by the renormalization group with an estimate for the power of the logarithmic factor of 0 2516 14 which is consistent with the predicted value of 1 4 we also characterize the behaviour of the pivot algorithm for sampling four dimensional self avoiding walks and conjecture that the probability of a pivot move being successful for an n step walk is o log n 1 4

finite range decomposition of gaussian processes let be the finite difference laplacian associated to the lattice z d for dimension d 3 a 0 and l a sufficiently large positive dyadic integer we prove that the integral kernel of the resolvent g a a 1 can be decomposed as an infinite sum of positive semi definite functions v n of finite range v n x y 0 for x y o l n equivalently the gaussian process on the lattice with covariance g a admits a decomposition into independent gaussian processes with finite range covariances for a 0 v n has a limiting scaling form l n left d 2 right gamma c left tfrac x y l n right as n as a corollary such decompositions also exist for fractional powers 2 0 2 the results of this paper give an alternative to the block spin renormalization group on the lattice

constructive field theory and applications perspectives and open problems in this paper we review many interesting open problems in mathematical physics which may be attacked with the help of tools from constructive field theory they could give work for future mathematical physicists trained with constructive methods well into the 21st century

self avoiding walk enumeration via the lace expansion we introduce a new method for the enumeration of self avoiding walks based on the lace expansion we also introduce an algorithmic improvement called the two step method for self avoiding walk enumeration problems we obtain significant extensions of existing series on the cubic and hypercubic lattices in all dimensions d 3 we enumerate 32 step self avoiding polygons in d 3 26 step self avoiding polygons in d 4 30 step self avoiding walks in d 3 and 24 step self avoiding walks and polygons in all dimensions d 4 we analyze these series to obtain estimates for the connective constant and various critical exponents and amplitudes in dimensions 3 d 8 we also provide major extensions of 1 d expansions for the connective constant and for two critical amplitudes

compressed self avoiding walks bridges and polygons we study various self avoiding walks saws which are constrained to lie in the upper half plane and are subjected to a compressive force this force is applied to the vertex or vertices of the walk located at the maximum distance above the boundary of the half space in the case of bridges this is the unique end point in the case of saws or self avoiding polygons this corresponds to all vertices of maximal height we first use the conjectured relation with the schramm loewner evolution to predict the form of the partition function including the values of the exponents and then we use series analysis to test these predictions

the intersection of brownian paths as a case study of a renormalization group method for quantum field theory a new approach is presented for the study of the probability that the random paths generated by two independent brownian motions in d intersect or more generally are within a short distance a of each other the well known behavior of that function of a above below and at the critical dimension d 4 as well as further corrections are derived here by means of a single renormalization group equation the equation s derivation is expected to shed some light on the function of the o d 4 quantum field theory

critical φ 4 3 e the euclidean 4 3 e model in r3 corresponds to a perturbation by a 4 interaction of a gaussian measure on scalar fields with a covariance depending on a real parameter e in the range 0 e 1 for e 1 one recovers the covariance of a massless scalar field in r3 for e 0 4 is a marginal interaction for 0 e 0 sufficiently small there exists a non gaussian fixed point with one unstable direction of the renormalization group iterations we construct the stable critical manifold near this fixed point and prove that under renormalization group iterations the critical theories converge to the fixed point

a supersymmetric transfer matrix and differentiability of the density of states in the one dimensional anderson model let h v on 2 z where v x 9 xez are d r v s with common probability distribution v let h t e itvdv v and let k e be the integrated density of states it is proven i if h is n times differentiable with h j t 0 l t for some 0 7 0 l n then k e is a cn function in particular if v has compact support and h t 0 l h 1 with 0 then k e is c this allows v to be singular continuous ii lfh t 0 e a tl for some 0 then k e is analytic in a strip about the real axis the proof uses the supersymmetric replica trick to rewrite the averaged green s function as a two point function of a one dimensional supersymmetric field theory which is studied by the transfer matrix method

renormalization group analysis of the self avoiding paths on the d dimensional sierpinski gaskets notion of the renormalization group dynamical system the self avoiding fixed point and the critical trajectory are mathematically defined for the set of self avoiding walks on the d dimensional pre sierpinski gaskets n simplex lattices such that their existence imply the asymptotic behaviors of the self avoiding walks such as the existence of the limit distributions of the scaled path lengths of canonical ensemble the connectivity constant exponential growth of path numbers with respect to the length and the exponent for mean square displacement we apply the so defined framework to prove these asymptotic behaviors of the restricted self avoiding walks on the 4 dimensional pre sierpinski gasket

construction and borel summability of infrared φ 4 4 by a phase space expansion we construct the thermodynamic limit of the critical massless 4 model in 4 dimensions with an ultraviolet cutoff by means of a partly renormalized phase space expansion this expansion requires in a natural way the introduction of effective or running constants and the infrared asymptotic freedom of the model i e the decay of the running coupling constant plays a crucial role we prove also that the correlation functions of the model are the borel sums of their perturbation expansion

the random walk representation of classical spin systems and correlation inequalities ferromagnetic lattice spin systems can be expressed as gases of random walks interacting via a soft core repulsion by using a mixed spinrandom walk representation we present a unified approach to many recently established correlation inequalities as an application of these inequalities we obtain a simple proof of the mass gap for the 4 2 quantum field model we also establish new upper bounds on critical temperatures

polymers in a weak random potential in dimension four rigorous renormalization group analysis correlation functions of the edwards model of polymers at weak coupling are defined and studied at the critical point in dimension four by a rigorous renormalization group method which validates at any order perturbative renormalization group results on their behaviour at large distances remainders are controlled by a new argument which enlarges the use of methods of constructive field theory to models of statistical physics

critical two point functions and the lace expansion for spread out high dimensional percolation and related models we consider spread out models of self avoiding walk bond percolation lattice trees and bond lattice animals on mathbb z d having long finite range connections above their upper critical dimensions d 4 self avoiding walk d 6 percolation and d 8 trees and animals the two point functions for these models are respectively the generating function for self avoiding walks from the origin to x in mathbb z d the probability of a connection from 0 to x and the generating function for lattice trees or lattice animals containing 0 and x we use the lace expansion to prove that for sufficiently spread out models above the upper critical dimension the two point function of each model decays at the critical point as a multiple of x 2 d as x to infty we use a new unified method to prove convergence of the lace expansion the method is based on x space methods rather than the fourier transform our results also yield unified and simplified proofs of the bubble condition for self avoiding walk the triangle condition for percolation and the square condition for lattice trees and lattice animals for sufficiently spread out models above the upper critical dimension

decay of correlations in nearest neighbor self avoiding walk percolation lattice trees and animals we consider nearest neighbor self avoiding walk bond percolation lattice trees and bond lattice animals on z d the two point functions of these models are respectively the generating function for self avoiding walks from the origin to x z d the probability of a connection from the origin to x and the generating functions for lattice trees or lattice animals containing the origin and x using the lace expansion we prove that the two point function at the critical point is asymptotic to const x 2 d as x for d 5 for self avoiding walk for d 19 for percolation and for sufficiently large d for lattice trees and animals these results are complementary to those of ann probab 31 2003 349 408 where spread out models were considered in the course of the proof we also provide a sufficient and rather sharp if d 4 condition under which the two point function of a random walk on z d is asymptotic to const x 2 d as x

critical polya urn we propose a variant model of the polya urn process where the dynamics consist of two competing elements suppression of growth and enhancement of dormant character here the level of such features is controlled by an internal parameter in a random manner so that the system self organizes and characteristic observables exhibit scale invariance suggesting the existence of criticality varying the internal control parameter one can explain interesting universality classes for avalanche type events we also discuss different versions of the model it is pointed out that such an underlying mechanism has earlier been noted to operate in a neural network

geometric analysis of φ4 fields and ising models parts i and ii we provide here the details of the proof announced in 1 that ind 4 dimensions the even 4 euclidean field theory with a lattice cut off is inevitably free in the continuum limit in the single phase regime the analysis is nonperturbative and is based on a representation of the field variables or spins in ising systems as source sink creation operators in a system of random currents which may be viewed as the mediators of correlations in this dual representation the onset of long range order is attributed to percolation in an ensemble of sourceless currents and the physical interaction in the 4 field and other aspects of the critical behavior in ising models are directly related to the intersection properties of long current clusters an insight into the criticality of the dimensiond 4 is derived from an analogy foreseen by k symanzik with the intersection properties of paths of brownian motion other results include the proof that in certain respect the critical behavior in ising models is in exact agreement with the mean field approximation in high dimensionsd 4 but not in the low dimensiond 2 for which we establish the universality of hyperscaling

weighted scale free networks with stochastic weight assignments we propose a model of weighted scale free networks incorporating a stochastic scheme for weight assignments to the links taking into account both the popularity and fitness of a node as the network grows the weights of links are driven either by the connectivity with probability p or by the fitness with probability 1 ensuremath p numerical results show that the total weight exhibits a power law distribution with an exponent ensuremath sigma that depends on the probability p the exponent ensuremath sigma decreases continuously as p increases for p 0 the scaling behavior is the same as that of the connectivity distribution an analytical expression for the total weight is derived so as to explain the features observed in the numerical results numerical results are also presented for a generalized model with a fitness dependent link formation mechanism

the optimal projection equations for fixed order dynamic compensation first order necessary conditions for quadratically optimal steady state fixed order dynamic compensation of a linear time invariant plant in the presence of disturbance and observation noise are derived in a new and highly simplified form in contrast to the pair of matrix riccati equations for the full order lqg case the optimal steady state fixed order dynamic compensator is characterized by four matrix equations two modified riccati equations and two modified lyapunov equations coupled by a projection whose rank is precisely equal to the order of the compensator and which determines the optimal compensator gains the coupling represents a graphic portrayal of the demise of the classical separation principle for the reduced order controller case

memory and burstiness in dynamic networks a discrete time random process is described which can generate bursty sequences of events a bernoulli process where the probability of an event occurring at time t is given by a fixed probability x is modified to include a memory effect where the event probability is increased proportionally to the number of events that occurred within a given amount of time preceding t for small values of x the interevent time distribution follows a power law with exponent 2 x we consider a dynamic network where each node forms and breaks connections according to this process the value of x for each node depends on the fitness distribution x from which it is drawn we find exact solutions for the expectation of the degree distribution for a variety of possible fitness distributions and for both cases where the memory effect either is or is not present this work can potentially lead to methods to uncover hidden fitness distributions from fast changing temporal network data such as online social communications and fmri scans

a renormalisation group method ii approximation by local polynomials this paper is the second in a series devoted to the development of a rigorous renormalisation group method for lattice field theories involving boson fields fermion fields or both the method is set within a normed algebra mathcal n of functionals of the fields in this paper we develop a general method localisation to approximate an element of mathcal n by a local polynomial in the fields from the point of view of the renormalisation group the construction of the local polynomial corresponding to f in mathcal n amounts to the extraction of the relevant and marginal parts of f we prove estimates relating f and its corresponding local polynomial in terms of the t phi semi norm introduced in part i of the series

the asymptotic evaluation of a class of path integrals ii the asymptotic behavior of a class of wiener like path integrals functions of the local time is determined these integrals are of interest in themselves and also arise very naturally in the theory of disordered systems we show that by making use of a grassman algebra i e a set of anticommuting variables the earlier treatment of this problem can be greatly simplified in particular the previous use of replica trick which involves a difficult to justify analytic continuation in the number of field components is thus avoided

next order asymptotic expansion for n marginal optimal transport with coulomb and riesz costs abstract motivated by a problem arising from density functional theory we provide the sharp next order asymptotics for a class of multimarginal optimal transport problems with cost given by singular long range pairwise interaction potentials more precisely we consider an n marginal optimal transport problem with n equal marginals supported on r d and with cost of the form i j x i x j s in this setting we determine the second order term in the n asymptotic expansion of the minimum energy for the long range interactions corresponding to all exponents 0 s d we also prove a small oscillations property for this second order energy term our results can be extended to a larger class of models than power law type radial costs such as non rotationally invariant costs the key ingredient and main novelty in our proofs is a robust extension and simplification of the fefferman gregg decomposition 20 26 extended here to our class of kernels and which provides a unified method valid across our full range of exponents our first result generalizes a recent work of lewin lieb and seiringer 36 who dealt with the second order term for the coulomb case s 1 d 3

asymptotic completeness and multiparticle structure in field theories the ideas developed in part i ref 1 are applied to the recently constructed massive gross neveu model we define in this case an irreducible kernel satisfying a regularized bethe salpeter equation which is convenient to derive asymptotic completeness in the 2 particle region as in part i the method allows direct graphical definition of general irreducible kernels and is well suited to the analysis of asymptotic completeness and related results in more general energy regions

self avoiding walk in five or more dimensions i the critical behaviour we use the lace expansion to study the standard self avoiding walk in thed dimensional hypercubic lattice ford 5 we prove that the numbercn ofn step self avoiding walks satisfiescn a n where is the connective constant i e 1 and that the mean square displacement is asymptotically linear in the number of steps i e v 1 2 a bound is obtained forcn x the number ofn step self avoiding walks ending atx the correlation length is shown to diverge asymptotically like z 1 2 the critical two point function is shown to decay at least as fast as x 2 and its fourier transform is shown to be asymptotic to a multiple ofk 2 ask 0 i e 0 we also prove that the scaling limit is gaussian in the sense of convergence in distribution to brownian motion the infinite self avoiding walk is constructed in this paper we prove these results assuming convergence of the lace expansion the convergence of the lace expansion is proved in a companion paper

growing random networks with fitness abstract three models of growing random networks with fitness dependent growth rates are analysed using the rate equations for the distribution of their connectivities in the first model a a network is built by connecting incoming nodes to nodes of connectivity k and random additive fitness with rate k 1 for 0 we find the connectivity distribution is power law with exponent 2 in the second model b the network is built by connecting nodes to nodes of connectivity k random additive fitness and random multiplicative fitness with rate k 1 this model also has a power law connectivity distribution but with an exponent which depends on the multiplicative fitness at each node in the third model c a directed graph is considered and is built by the addition of nodes and the creation of links a node with fitness i incoming links and j outgoing links gains a new incoming link with rate i 1 and a new outgoing link with rate j 1 the distributions of the number of incoming and outgoing links both scale as power laws with inverse logarithmic corrections

a renormalizable field theory the massive gross neveu model in two dimensions the euclidean massive gross neveu model in two dimensions is just renormalizable and asymptotically free thanks to the pauli principle bare perturbation theory with an ultra violet cut off and the correct ansatz for the bare mass is convergent in a disk whose radius corresponds by asymptotic freedom to a small finite renormalized coupling constant therefore the theory can be fully constructed in a perturbative way it satisfies the o s axioms and is the borel sum of the renormalized perturbation expansion of the model

logarithmic correction for the susceptibility of the 4 dimensional weakly self avoiding walk a renormalisation group analysis we prove that the susceptibility of the continuous time weakly self avoiding walk on mathbb z d in the critical dimension d 4 has a logarithmic correction to mean field scaling behaviour as the critical point is approached with exponent frac 1 4 for the logarithm the susceptibility has been well understood previously for dimensions d 5 using the lace expansion but the lace expansion does not apply when d 4 the proof begins by rewriting the walk two point function as the two point function of a supersymmetric field theory the field theory is then analysed via a rigorous renormalisation group method developed in a companion series of papers by providing a setting where the methods of the companion papers are applied together the proof also serves as an example of how to assemble the various ingredients of the general renormalisation group method in a coordinated manner