on the statistical mechanics of noncrossing chains part 3 for pt 2 see ibid vol 22 p l843 1989 the author has shown in a previous paper that the generating function for self avoiding walks with end to end displacement specified can be converted into a mayer like cluster series by simply taking the reciprocal of the generating function in a previous letter the results of doing this for the plane square lattice were reported for terms up to z16 the cluster series terms are much less numerous and numerically much smaller than those of the raw self avoiding walk series and point to the existence of a set of rapidly converging successive approximations for such problems guttman s conclusion that the singularity of the self avoiding walk series may be a confluent one received support he now reports the results of extending the work on the plane square lattice up to z25 and the results of similar work on the simple cubic lattice up to z15 and discusses them briefly conclusions from the earlier work are confirmed

gradient systems in view of information geometry abstract dualistic properties of a gradient flow on a manifold m associated with a dualistic structure g is studied from an information geometrical viewpoint some useful applications are also investigated

invariance properties in hermite pade approximation theory abstract the pade approximant is invariant under both linear fractional transformations of the function value and linear fractional transformations origin fixed of the argument the paper reports on an investigation of these invariance properties to the class of hermite pade approximants based on the class of general ordinary nonlinear differential equations further the paper characterizes the class with these invariance properties uniqueness is discussed examples are given such as the pade riccati approximants and allowed types of singular points are discussed

connective constant of the self avoiding walk on the triangular lattice the author uses a recently developed monte carlo method to study the connective constant of the self avoiding walk saw on the triangular lattice assuming gamma 43 32 one finds mu t 4 15096 or 0 00010 or 0 00026 95 confidence limits and mu t 4 15096 or 0 00020 or 0 00037 99 confidence limits where the first error bar represents systematic error due to corrections to scaling and the second error bar represents statistical error this rules out the conjecture mu t mu h 6 at the greater than 99 confidence level

the kazhdan lusztig basis and the temperley lieb quotient in type d let h be a hecke algebra associated with a coxeter system of type d and let tl be the corresponding temperley lieb quotient the algebra tl admits a canonical basis which facilitates the construction of irreducible representations in this paper we explain the relationship between the canonical basis of tl and the kazhdan lusztig basis of h

divergence from factorizable distributions and matroid representations by partitions maximization of the information divergence from any hierarchical log linear model is studied a new upper bound on the maximum is presented and its tightness analyzed for the models given by the bases of a matroid the latter is related to matroid representations by partitions or equivalently to ideal secret sharing schemes a new link between the divergence maximization the maximum likelihood principle and secret sharing is established

monte carlo studies of three dimensional two step restricted self avoiding walks two step restricted walk tsrw models are a class of restricted self avoiding walk saw where in addition to the self avoidance constraint certain restrictions are placed upon each pair of successive steps in this paper we explore the relationship between the restrictions and the scaling of the average size of walks in three dimensional models we use the pruned enriched rosenbluth method algorithm to perform monte carlo studies in five representative tsrw models in three dimensions the results present strong numerical evidence that all non trivial tsrw models in three dimensions have the same size scaling behaviour as unrestricted saws this is in contrast to two dimensions where several universality classes are accepted to exist in particular we find no rule analogous to the spiral walk of two dimensions

correction to scaling exponents and amplitudes for the correlation length of linear polymers in two dimensions we consider the scaling behaviour of the radius of gyration for a system of dilute linear polymers we focus on pn the mean square end to end distance of an n step self avoiding walk for which an additional two terms were recently calculated for the close packed triangular lattice combining several extrapolation methods we find that a consistent description of the scaling behaviour exists if and only if the correction to scaling exponent a is roughly half as large as commonly believed we conclude that all data ale consistent with the equation pn an2 1 b na c n where y a a 1 j2 abso 21 and c lo

information geometry on hierarchy of probability distributions an exponential family or mixture family of probability distributions has a natural hierarchical structure this paper gives an orthogonal decomposition of such a system based on information geometry a typical example is the decomposition of stochastic dependency among a number of random variables in general they have a complex structure of dependencies pairwise dependency is easily represented by correlation but it is more difficult to measure effects of pure triplewise or higher order interactions dependencies among these variables stochastic dependency is decomposed quantitatively into an orthogonal sum of pairwise triplewise and further higher order dependencies this gives a new invariant decomposition of joint entropy this problem is important for extracting intrinsic interactions in firing patterns of an ensemble of neurons and for estimating its functional connections the orthogonal decomposition is given in a wide class of hierarchical structures including both exponential and mixture families as an example we decompose the dependency in a higher order markov chain into a sum of those in various lower order markov chains

a diagram calculus for certain canonical bases we introduce a certain cellular algebra which is a quotient of the q schur algebra this is naturally equipped with a canonical basis which is compatible with lusztig s canonical bases for certain modules for the quantized enveloping algebra we describe a diagram calculus for which makes calculations involving the corresponding canonical bases easy to understand

critical exponents for the self avoiding random walk in three dimensions we compute by direct monte carlo simulation the main critical exponents 4 andv and the effective coordination number for the self avoiding random walk in three dimensions on a cubic lattice we find both hyperscaling relationsdv 2 anddv 2 4 0 satisfied ind 3

self avoiding polygons on the square l and manhattan lattices transfer matrix techniques are used to extend the self avoiding polygon generating function on the square lattice to terms in x46 corresponding to 46 step polygons these techniques are then extended to apply to directed square lattices such as the l and manhattan lattice and the self avoiding polygon generating function to x48 is found for these lattices series analysis confirms that the specific heat exponent alpha 1 2 for the self avoiding walk problem and gives the following estimates for the connective constants mu sq 2 638155 or 0 000025 mu l 1 5657 or 0 0019 and mu man 1 7328 or 0 0005 some evidence for a correction to scaling exponent delta approximately 0 84 is found from square lattice series

universal distance ratios for 2d saws series results the mean square distance of a monomer to the origin and the mean square distance of the centre of mass to the origin are studied by exact enumeration on the square and triangular lattices the numerical results are inconsistent with the theoretical prediction of cardy and saleur 1989 by more than fifteen times the numerical uncertainty in the authors calculation

position space renormalisation group for directed diffusion limited aggregation the author uses a position space renormalisation group transformation to show that directed diffusion limited aggregation and directed animals are in different universality classes

q schur algebras asymptotic forms and quantum sln abstract we studied asymptotic methods for q schur algebras and related quantum groups and constructed a kind of asymptotic form for q schur algebras and quantum special linear groups the structure and representations for these forms were also discussed

universal amplitude ratios for three dimensional self avoiding walks we have calculated exactly the number the mean square end to end distance the mean square radius of gyration and the mean square distance of a monomer from the origin for n step self avoiding walks on the simple cubic diamond body centred cubic and face centred cubic lattices respectively up to 20 30 16 and 13 steps by a computer two universal amplitude ratios are estimated

monomials and temperley lieb algebras we classify the fully tight simply laced coxeter groups that is the ones whoseiji avoiding kazhdan lusztig basis elements are monomials in the generatorsbsi we then investigate the basis of the temperley lieb algebra arising from the kazhdan lusztig basis of the associated hecke algebra and prove that the basis coincides with the usual monomial basis

multi information in the thermodynamic limit a multivariate generalization of mutual information multi information is defined in the thermodynamic limit the definition takes phase coexistence into account by taking the infimum over the translation invariant gibbs measures of an interaction potential it is shown that this infimum is attained in a pure state an explicit formula can be found for the ising square lattice where the quantity is proved to be maximized at the phase transition point by this phase coexistence is linked to high model complexity in a rigorous way

on three dimensional self avoiding walk symmetry classes in two dimensions the universality classes of self avoiding walks saws on the square lattice restricted by allowing only certain two step configurations tscs to occur within each walk has been argued to be determined primarily by the symmetry of the set of allowed rules in three dimensions early work tentatively found one undirected universality class different to that of unrestricted saws on the simple cubic lattice this rule was a natural generalization of the square lattice spiral saw to three dimensions in this report we examine a variety of three dimensional saw models with different step restrictions carefully chosen so as to search for a connection between the symmetry of the rules and possible new universality classes a first analysis of the scaling of the radius of gyration suggests several universality classes including the one found earlier and perhaps some novel class es however a classification of these universality classes using the symmetries of the rules or other basic rule properties is not evident further analysis of the number of configurations and moment of inertia tensor suggests that in three dimensions the only non trivial or undirected universality class is that of unrestricted saws

knot localization in adsorbing polymer rings we study by monte carlo simulations a model of a knotted polymer ring adsorbing onto an impenetrable attractive wall the polymer is described by a self avoiding polygon on the cubic lattice we find that the adsorption transition temperature the crossover exponent ensuremath phi and the metric exponent ensuremath nu are the same as in the model where the topology of the ring is unrestricted by measuring the average length of the knotted portion of the ring we are able to show that adsorbed knots are localized this knot localization transition is triggered by the adsorption transition but is accompanied by a less sharp variation of the exponent related to the degree of localization indeed for a whole interval below the adsorption transition one can not exclude a contiuous variation with temperature of this exponent deep into the adsorbed phase we are able to verify that knot localization is strong and well described in terms of the flat knot model

on the entropy of couplings abstract in this paper some general properties of shannon information measures are investigated over sets of probability distributions with restricted marginals certain optimization problems associated with these functionals are shown to be np hard and their special cases are found to be essentially information theoretic restatements of well known computational problems such as the subset sum and the 3 partition the notion of minimum entropy coupling is introduced and its relevance is demonstrated in information theoretic computational and statistical contexts finally a family of pseudometrics on the space of discrete probability distributions defined by these couplings is studied in particular their relation to the total variation distance and a new characterization of the conditional entropy is given

composite models of polygons a composite polygon is composed of a lattice polygon in the square lattice which contains in its interior an internal structure which may also be a lattice polygon or a lattice tree or a lattice animal or a lattice disc or a collection of these the properties of composite polygons are considered in this manuscript in particular i shall consider the growth constants and generating functions of these models as well as the statistical mechanics of interacting models of composite polygons it is shown that there is an adsorption transition of the internal structure on the containing polygon and a transition which corresponds to the inflation of the containing polygon by the internal structure

structure of a hecke algebra quotient let w be a coxeter group with coxeter graph f let 7h be the associated hecke algebra we define a certain ideal i in 7h and study the quotient algebra 7h 7 i we show that when f is one of the infinite series of graphs of type e the quotient is semi simple we examine the cell structures of these algebras and construct their irreducible representations we discuss the case where f is of type b f or h department of mathematics harvard university cambridge massachusetts 02138 e mail address ckf anfmath harvard edu current address school of mathematics institute for advanced study princeton new jersey 08540 this content downloaded from 207 46 13 105 on wed 25 may 2016 06 46 50 utc all use subject to http about jstor org terms

kazhdan lusztig immanants and products of matrix minors abstract using the kazhdan lusztig basis c w 1 w s n for the symmetric group algebra we obtain nonnegativity properties of certain polynomials in matrix minors in particular we show that the application of these polynomials to jacobi trudi matrices yields symmetric functions which are equal to nonnegative linear combinations of schur functions

critical exponents hyperscaling and universal amplitude ratios for two and three dimensional self avoiding walks we make a high precision monte carlo study of two and three dimensional self avoiding walks saws of length up to 80 000 steps using the pivot algorithm and the karp luby algorithm we study the critical exponents v and 2 triangle sub 4 gamma as well as several universal amplitude ratios in particular we make an extremely sensitive test of the hyperscaling relation dv 2 triangle sub 4 gamma in two dimensions we confirm the predicted exponent v 3 4 and the hyperscaling relation we estimate the universal ratios 0 14026 0 00007 0 43961 0 00034 and psi 0 66296 0 00043 68 confidence limits in three dimensions we estimate v 0 5877 0 0006 with a correction to scaling exponent triangle sub 1 0 56 0 03 subjective 68 confidence limits this value for v agrees excellently with the field theoretic renormalization group prediction but there is some discrepancy for triangle 1 earlier monte carlo estimates of v which were approximately 0 592 are now seen to be biased by corrections to scaling we estimate the universal ratios 0 1599 0 0002 and psi 0 2471 0 0003 since psi 0 hyperscaling holds the approach to psi is from above contrary to the prediction of the two parameter renormalization group theory we critically reexamine this theory and explain where the error lies in an appendix we prove rigorously modulo some standard scalingmore assumptions the hyperscaling relation dv 2 triangle sub 4 gamma for two dimensional saws less

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

theory of self avoiding walks on percolation fractals a phenomenological approach which takes into account the basic geometry and topology of percolation fractal structures and of self avoiding walks saw is used to derive a new expression for the flory exponent describing the average radius of gyration of saws on fractals we focus on the radius of gyration and discuss the importance of the intrinsic fractal dimensions of percolation clusters in determining the lower and upper critical dimensions of saws the mean field version of our new formula corresponds to the aharony and harris expression who used the standard flory approach for its derivation

quantum convex support abstract convex support the mean values of a set of random variables is central in information theory and statistics equally central in quantum information theory are mean values of a set of observables in a finite dimensional c algebra a which we call quantum convex support the convex support can be viewed as a projection of the state space of a and it is a projection of a spectrahedron spectrahedra are increasingly investigated at least since the 1990s boom in semi definite programming we recall the geometry of the positive semi definite cone and of the state space we write a convex duality for general self dual convex cones this restricts to projections of state spaces and connects them to results on spectrahedra our main result is an analysis of the face lattice of convex support by mapping this lattice to a lattice of orthogonal projections using natural isomorphisms the result encodes the face lattice of the convex support into a set of projections in a and enables the integration of convex geometry with matrix calculus or algebraic techniques

dual braid monoids mikado braids and positivity in hecke algebras we study the rational permutation braids that is the elements of an artin tits group of spherical type which can be written x 1 y where x and y are prefixes of the garside element of the braid monoid we give a geometric characterization of these braids in type a n and b n and then show that in spherical types different from d n the simple elements of the dual braid monoid for arbitrary choice of coxeter element embedded in the braid group are rational permutation braids we conjecture this to hold also in type d n this property implies positivity properties of the polynomials arising in the linear expansion of their images in the iwahori hecke algebra when expressed in the kazhdan lusztig basis in type a n it implies positivity properties of their images in the temperley lieb algebra when expressed in the diagram basis

the high temperature susceptibility and spin spin correlation function of the three dimensional ising model the series analysis protocol developed in a preceding paper ibid vol 20 p 1839 1987 is applied to the susceptibility and correlation function of the three dimensional ising model for the spin 1 2 ising data the author argues that the protocol needs modification to take account of trends in the data which trends arise from the nature of the series in this way the author obtains gamma 1 239 or 0 003 for all cubic lattices and for a variety of spin values the diamond lattice data lies outside this range and it is argued that this series is effectively far shorter than the others the author also estimates nu 0 632 0 002 0 003 for the bcc lattice and obtains critical point estimates for all lattice and spin models studied

on the critical behaviour of self avoiding walks for pt i see ibid vol 22 p 1839 1989 for the self avoiding walk problem the coefficients of the chain generating function and of the generating function for the sum of square end to end distances have been extended to 20 terms for the triangular lattice to 27 terms for the diamond lattice to 21 terms for the simple cubic lattice and to 16 terms for the bcc lattice precise estimates of the critical points are obtained and for the exponents we find that gamma 1 161 or 0 002 and nu 0 592 or 0 003 encompasses all the three dimensional lattice data

critical exponents of the self avoiding walks on a family of finitely ramified fractals the authors have studied the self avoiding walks saw on a family of finitely ramified fractals the first member b 2 of the family is the two dimensional sierpinski gasket while the last member b infinity appears to be a wedge of the homogeneous triangular lattice by means of the exact renormalisation group transformations they have calculated the critical exponents alpha nu and gamma and the connectivity constant mu of saw on each member of a sequence 2 or b or 8 of the studied fractal family the obtained exact results are compared with the recent phenomenological proposals and with the results believed to be exact in the case of a homogeneous two dimensional lattice

connective constant of saws on the sierpinski gasket family using a graph counting technique suitable for regular fractals an exact evaluation of the total number of embeddings of self avoiding walks on the generalized sierpinski gasket is obtained numerical estimates for the connective constants are quoted for the first time where b is the generation parameter of the gaskets it is shown that the number of distinct n step saws per site converges to the triangular lattice values when our analysis indicates that converges to the euclidean value in the same limit and an asymptotic expression is given

critical behavior of self avoiding walks on fractals using a new graph counting technique suitable for self similar fractals exact 18th order series expansions for saws on some sierpinski carpets are generated from them the critical fugacityxc and critical exponents saw and saw are obtained the results show a linear dependence of the critical fugacity with the average number of bonds per site of the lattices studied we find for some carpets with low lacunarity that saw saw d for saws on the fractals which are embedded in ad dimensional euclidean space

the extension of self avoiding random walk series in two dimensions the authors have extended the enumeration of square lattice and triangular lattice self avoiding walks and their end to end distance by two terms they have also calculated 17 terms in the series for the resistance of saws with bridges on the square lattice analysis of the new data allows the connective constants to be more accurately estimated and affirms the values gamma 43 32 and v 3 4 and 1 gamma for the exponent of the resistance series

continuity of the maximum entropy inference we study the inverse problem of inferring the state of a finite level quantum system from expected values of a fixed set of observables by maximizing a continuous ranking function we have proved earlier that the maximum entropy inference can be a discontinuous map from the convex set of expected values to the convex set of states because the image contains states of reduced support while this map restricts to a smooth parametrization of a gibbsian family of fully supported states here we prove for arbitrary ranking functions that the inference is continuous up to boundary points this follows from a continuity condition in terms of the openness of the restricted linear map from states to their expected values the openness condition shows also that ranking functions with a discontinuous inference are typical moreover it shows that the inference is continuous in the restriction to any polytope which implies that a discontinuity belongs to the quantum domain of non commutative observables and that a geodesic closure of a gibbsian family equals the set of maximum entropy states we discuss eight descriptions of the set of maximum entropy states with proofs of accuracy and an analysis of deviations

entropy distance new quantum phenomena we study a curve of gibbsian families of complex 3 3 matrices and point out new features absent in commutative finite dimensional algebras a discontinuous maximum entropy inference a discontinuous entropy distance and non exposed faces of the mean value set we analyze these problems from various aspects including convex geometry topology and information geometry this research is motivated by a theory of infomax principles where we contribute by computing first order optimality conditions of the entropy distance

random walks with short range interaction and mean field behavior we introduce a model of self repelling random walks where the short range interaction between two elements of the chain decreases as a power of the difference in proper time the model interpolates between the lattice edwards model and the ordinary random walk we show by means of monte carlo simulations in two dimensions that the exponentvmf obtained through a mean field approximation correctly describes the numerical data and is probably exact as long as it is smaller than the corresponding exponent for self avoiding walks we also compute the exponent and present a numerical study of the scaling functions

an information geometric approach to a theory of pragmatic structuring 1 1 the motivation of the approach in the field of neural networks so called infomax principles like the principle of maximum information preservation by linsker 20 are formulated to derive learning rules that improve the information processing properties of neural systems see 12 these principles which are based on information theoretic measures are intended to describe the mechanism of learning in the brain there the starting point is a low dimensional and biophysiologically motivated parametrization of the neural system which need not necessarily be compatible with the given optimization principle in contrast to this we establish theoretical results about the low complexity of optimal solutions for the optimization problem of frequently used measures like the mutual information in an unconstrained and more theoretical setting in the present paper we do not comment on applications to modeling neural networks this is intended to be done in a further step where the results can be used for the characterization of good parameter sets that on the one hand are compatible with the underlying optimization and on the other hand are biologically motivated

parisi s mean field solution for spin glasses as an analytic continuation in the replica number parisi s replica symmetry breaking solution for spin glasses is extended to finite replica number n the free energy fp n obtained this way as well as its first two derivatives with respect to n are shown to join the corresponding values in the sherrington kirkpatrick sk solution at a characteristic value ns t where stability breaks down in the latter the continuation composed of the sk branch fsk n for n or ns t and the parisi branch fp n for 0 or n or ns t fulfils the requirements of convexity monotonicity and stability for all n

critical exponent for the loop erased self avoiding walk by monte carlo methods a monte carlo simulation was performed for loop erased self avoiding walks lesaw to ascertain the exponentv for the z2 and z3 lattices the estimated values were 2v 1 600 0 006 in two dimensions and 2v 1 232 0 008 in three dimensions leading to the conjecturev 4 5 for the two dimensional lesaw these results add to existing evidence that the loop erased self avoiding walks are not in the same universality class as self avoiding walks

nonergodicity of local length conserving monte carlo algorithms for the self avoiding walk it is proved that every dynamic monte carlo algorithm for the self avoiding walk based on a finite repertoire of local n conserving elementary moves is nonergodic heren is the number of bonds in the walk indeed for largen each ergodic class forms an exponentially small fraction of the whole space this invalidates at least in principle the use of the verdier stockmayer algorithm and its generalizations for high precision monte carlo studies of the self avoiding walk

the theory of angular momenta and higher su n symmetries the authors discuss interrelations between generating invariants gi for the reduction pi mi 1 d pi to d on 1 of the m fold tensor product of su n irreducible representations irreps d pi and polynominal bases of the su m irreps d on 1j om n 1 ok 0 0 0 k times a realisation of the su m irrep d oj om 3 bases is given in terms of gi for su 2 coupling wigner coefficients as a byproduct an expression is obtained for su 2 6 j symbols in terms of only two wigner coefficients the authors also discuss some possibilities of the analysis involved in solving the wigner biedenharn problem construction of orthonormal sets of the wigner coefficients for su n groups n or 3

criticality of self avoiding walks with an excluded infinite needle the authors study the effect of repulsion for self avoiding walks and random walks from excluded sets they show in particular that the mean displacement away from an excluded infinite needle of self avoiding random walks in three dimensions has to diverge along the privileged axis as nsigma where n is the number of steps and sigma is a sub leading critical exponent for the two point function this exponent has been determined by using a high precision monte carlo simulation sigma 0 370 or 0 011 its knowledge is used to improve the measure of universal quantities like the exponent nu nu 0 5867 or 0 0025 in agreement with the in expansion estimate and with experimental data and amplitude ratios they verify also that for simple random walks the excluded needle introduces instead logarithmic violations to scaling

schur positivity and schur log concavity we prove okounkov s conjecture a conjecture of fomin fulton li poon and a special case of lascoux leclerc thibon s conjecture on schur positivity and give several more general statements using a recent result of rhoades and skandera we also give an intriguing log concavity property of schur functions

join and cut algorithm for self avoiding walks with variable length and free endpoints we introduce a new monte carlo algorithm for generating self avoiding walks of variable length and free endpoints the algorithm works in the unorthodox ensemble consisting of all pairs of saws such that the total number of stepsntot in the two walks is fixed the elementary moves of the algorithm are fixed n e g pivot moves on the individual walks and a novel join and cut move that concatenates the two walks and then cuts them at a random location we analyze the dynamic critical behavior of the new algorithm using a combination of rigorous heuristic and numerical methods in two dimensions the autocorrelation time in cpu units grows as n 1 5 and the behavior improves in higher dimensions this algorithm allows high precision estimation of the critical exponent

the dimensions of knotted polygons the authors study the dimensions mean square radius of gyration and mean span of self avoiding polygons on the simple cubic lattice with fixed knot type the approach used is a monte carlo algorithm which is a combination of the bfacf algorithm and the pivot algorithm so that the polygons are studied in the grand canonical ensemble but the autocorrelation time is not too large they show that although the dimensions of polygons are sensitive to knot type the critical exponent v and the leading amplitude are independent of the knot type of the polygon the knot type influences the confluent correction to the scaling term and hence the rate of approach to the limiting behaviour

repulsion of random and self avoiding walks from excluded points and lines the authors investigate the repulsion of random walks rws and self avoiding walks saws induced by a excluding a single lattice point or b excluding all points on a half line from infinity to the origin for saws they use exact enumeration and monte carlo methods to study the asymptotic behaviour of the displacement away from the excluded set on three four and five dimensional hypercubic lattices when the saw begins one lattice site away from an excluded point along the x direction the mean displacement after n steps xn approaches a finite limit at a power law rate as n to infinity however the distribution of projected displacements exhibits a residual asymmetry in the asymptotic limit reflective of a long range influence of the excluded point this general pattern of behaviour also occurs for a purely random walk for a saw which starts one lattice spacing away from the axis of the excluded half line enumeration data suggest that the mean displacement along the axis diverges as n nu with v 3 4 in two dimensions but as nz with z approximately 0 35 in three dimensions this unexpected behaviour appears to be corroborated by constant fugacity monte carlo data however for the corresponding rw model both the solution to the diffusion equation and a heuristic argument indicate that xn diverges as n1 2 in two dimensions and as n1 2 ln n in three dimensions

unitary representations branching rules and matrix elements for the non compact symplectic groups the complementarity of the symplectic and orthogonal groups is used to infer properties of the infinite dimensional unirreps of the former from the character theory of the latter the complete set of d series metaplectic unirreps of sp n r is identified and branching rules are given for their restrictions to the maximal compact subgroup u n developed in terms of the properties of schur functions a known algorithm for the evaluation of matrix elements of the sp 3 r lie algebra is extended to any sp n r and analytic expressions are given for important classes of unirreps and multiplicity free states

monte carlo simulation of 3d self avoiding walks presents simulation results for 3d self avoiding walks of moderate length n or 2000 using a recently introduced recursive sampling method it is shown that this method is indeed maximally efficient for determining the connectivity constant mu combining the data with previous results from exact enumerations and from simulations of very long walks using the pivot algorithm the author obtains very precise estimates of mu and of the critical exponents gamma and nu he also points out significant deviations from uniformity in a very popular random number generator which was considered safe until now

an extension of the two dimensional self avoiding walk series on the square lattice the authors have adapted the dimerization algorithm for enumerating self avoiding walks to exploit the parallelism of the connection machine system by this approach they have extended the series for the number of self avoiding walks on the 2d square lattice by five terms these data may permit more accurate estimates of critical properties

locality of global stochastic interaction in directed acyclic networks the hypothesis of invariant maximization of interaction imi is formulated within the setting of random fields according to this hypothesis learning processes maximize the stochastic interaction of the neurons subject to constraints we consider the extrinsic constraint in terms of a fixed input distribution on the periphery of the network our main intrinsic constraint is given by a directed acyclic network structure first mathematical results about the strong relation of the local information flow and the global interaction are stated in order to investigate the possibility of controlling imi optimization in a completely local way furthermore we discuss some relations of this approach to the optimization according to linsker s infomax principle

on planar algebras arising from hypergroups abstract let a be an associative algebra with identity and with trace we study the family of planar algebras on 1 boxes that arise from a in the work of jones but with the added assumption that the labels on the 1 boxes come from a discrete hypergroup in the sense of sunder this construction equips the algebra pna with a canonical basis b n a which turns out to be a tabular basis we examine special cases of this construction to exhibit a close connection between such bases and kazhdan lusztig bases of hecke algebras of types a b h or i

categorification of the temperley lieb algebra by bimodules abstract we categorify the temperley lieb algebra by analogues of soergel bimodules the key point is that the product of bimodules is not given by a usual tensor product but by a slightly more complicated operation

information geometry and sufficient statistics information geometry provides a geometric approach to families of statistical models the key geometric structures are the fisher quadratic form and the amari chentsov tensor in statistics the notion of sufficient statistic expresses the criterion for passing from one model to another without loss of information this leads to the question how the geometric structures behave under such sufficient statistics while this is well studied in the finite sample size case in the infinite case we encounter technical problems concerning the appropriate topologies here we introduce notions of parametrized measure models and tensor fields on them that exhibit the right behavior under statistical transformations within this framework we can then handle the topological issues and show that the fisher metric and the amari chentsov tensor on statistical models in the class of symmetric 2 tensor fields and 3 tensor fields can be uniquely up to a constant characterized by their invariance under sufficient statistics thereby achieving a full generalization of the original result of chentsov to infinite sample sizes more generally we decompose markov morphisms between statistical models in terms of statistics in particular a monotonicity result for the fisher information naturally follows

end to end distance of linear polymers in two dimensions a reassessment recent exact enumeration studies of the two dimensional self avoiding walk have produced evidence in support of non analytic scaling corrections newly extended series for the triangular and square lattices to n 19 and 25 respectively are examined in the light of recent monte carlo results suggesting the absence of corrections of this type using a modified approach to series analysis it is concluded that the end to end distance can best be understood in terms of analytic corrections alone in view of the lack of theoretical support there is little reason to introduce additional kinds of correction in attempting to explain the data

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

the self avoiding walk on the honeycomb lattice the author analyses the chain generating function for the self avoiding walk problem on the honeycomb lattice she uses the exact critical point value of nienhuis and finds exponent values of gamma approximately 1 344 and delta 1 approximately 0 93 for the dominant and first confluent exponents the former value is in agreement with nienhuis exact result second and third confluent exponents delta 2 approximately 1 2 and delta 3 approximately 1 5 are also identified these exponents being consistent with monte carlo results and nienhuis exact irrelevant eigenvalue respectively

new upper bounds for the connective constants of self avoiding walks using a novel implementation of the goulden jackson method we compute new rigorous upper bounds for the connective constants of self avoiding walks breaking alm s previous records for rectangular hypercubic lattices we also give the explicit generating functions for memory 8 we then incorporate a numerical limit which gives bounds that are even better

knotting probabilities after a local strand passage in unknotted self avoiding polygons we investigate both theoretically and numerically the knotting probabilities after a local strand passage is performed in an unknotted self avoiding polygon sap on the simple cubic lattice in the polygons studied it is assumed that two polygon segments have already been brought close together for the purpose of performing a strand passage this restricts the polygons considered to those that contain a specific pattern called at a fixed location an unknotted polygon containing is called a sap it is proved that the number of n edge saps grows exponentially with n at the same rate as the total number of n edge unknotted saps those with no prespecified strand passage structure furthermore it is proved that the same holds for subsets of n edge saps that yield a specific after strand passage knot type thus the probability of a given after strand passage knot type does not grow or decay exponentially with n instead it is conjectured that these after strand passage knot probabilities approach as n goes to infinity knot type dependent amplitude ratios lying strictly between 0 and 1 this conjecture is supported by numerical evidence from monte carlo data generated using a composite aka multiple markov chain monte carlo bfacf algorithm developed to study saps a new maximum likelihood method is used to estimate the critical exponents relevant to this conjecture we also obtain strong numerical evidence that the after strand passage knotting probability depends on the local structure around the strand passage site if the local structure and the crossing sign at the strand passage site are considered then we observe that the more compact the local structure the less likely the after strand passage polygon is to be knotted this trend for compactness versus knotting probability is consistent with results obtained for other strand passage models however we are the first to note the influence of the crossing sign information we use two measures of compactness one involves the size of a smallest polygon that contains the structure and the other is in terms of an opening angle the opening angle definition is consistent with one that is measurable from single molecule dna experiments the theoretical and numerical approaches presented here are more broadly applicable to other sap models

self avoiding walk connectivity constant and theta point on percolating lattices the average connectivity constant mu of self avoiding walks saws is obtained from exact enumeration of saws on monte carlo generated percolating clusters in a randomly diluted square lattice for averages over the infinite percolating cluster mu decreases almost linearly with bond dilution 1 p where p is the bond occupation concentration the authors find mu pc 1 31 or 0 03 at the percolation threshold pc and could not detect any significant difference between mu pc and pc mu 1 the variation of theta point for saws on the same lattice with dilution is also estimated analysing the partition function zeros within the limited accuracy of their analysis its variation with dilution is observed as being quite weak and the theta point increases somewhat compared to pure lattice value near pc they find a non vanishing theta point ktheta pc equivalent to 0 59 where k0 j k theta on the square lattice percolation cluster at pc

convergence and extrapolation in finite size scaling renormalization abstract finite size renormalization technique is studied by examining a test model which is exactly solvable and the problem of lattice animals and self avoiding walks bulk estimates of the leading irrelevant rg eigenvalue are found and compared with independent estimates from finite size data we introduce an extrapolation technique which is applicable to non monotonic sequences of exponent approximants

tests of hyperuniversality for self avoiding walks a universal combination of amplitudes for the end to end distance of n step self avoiding walks and for the number of n step self avoiding polygons is estimated numerically for various lattices in d 2 universality of this amplitude combination is confirmed to good accuracy while in d 3 the data are consistent with universality but the error limits are rather large for finite width square lattice strips the authors calculate universal critical point correlation amplitudes by fixed fugacity monte carlo simulations in addition these amplitudes are calculated analytically either exactly or approximately by cardy s conformal mapping approach numerical values agree well with the theoretical predictions

on three dimensional self avoiding walks long self avoiding walks of up to 2400 steps have been generated on the sc and bcc lattices using an improved monte carlo technique analysis of the asymptotic length dependence of the end to end distance and radius of gyration of the walks leads to values in the range 0 591 0 593 for the critical exponent nu the simple power law dependence on length provides an excellent fit to the data over walk lengths between 120 and 2400 with fluctuations around the asymptotic results averaging to a mere 0 13 systematic deviations from asymptotic behaviour in short self avoiding walks have also been examined using monte carlo generated walks ranging in length from 12 to 60 the results do not support the existence of the non analytic correction predicted by the renormalisation group in the light of this unexpected result available series expansions for walks on the fcc lattice have been re examined and previous claims to have observed non analytic behaviour questioned equal if not better convergence of the extrapolated series can be obtained without resorting to non analytic correction terms finally an analysis has been made of the radius of gyration series to which several new terms have been added

two dimensional polymers universality and correction to scaling improved monte carlo methods have been used to study the asymptotic behaviour of self avoiding walks on two dimensional lattices the mean square end to end distance rn2 and radius of gyration sn2 are both found to scale as n2v with exponent values very close to the expected v 3 4 and an amplitude ratio that is universal small n deviations of rn2 from the asymptotic results indicate a correction to scaling exponent delta 1 a value that differs from previous estimates

a new algorithm to enumerate the self avoiding random walk presents a new algorithm for enumerating the self avoiding walk and extend the existing series of 19 terms by one more term on a triangular lattice the leading correction to scaling exponent delta 1 for the mean square end to end distance of the self avoiding walk is found to be less than one

the pivot algorithm a highly efficient monte carlo method for the self avoiding walk the pivot algorithm is a dynamic monte carlo algorithm first invented by lal which generates self avoiding walks saws in a canonical fixed n ensemble with free endpoints heren is the number of steps in the walk we find that the pivot algorithm is extraordinarily efficient one effectively independent sample can be produced in a computer time of ordern this paper is a comprehensive study of the pivot algorithm including a heuristic and numerical analysis of the acceptance fraction and autocorrelation time an exact analysis of the pivot algorithm for ordinary random walk a discussion of data structures and computational complexity a rigorous proof of ergodicity and numerical results on self avoiding walks in two and three dimensions our estimates for critical exponents are 0 7496 0 0007 ind 2 and 0 592 0 003 ind 3 95 confidence limits based on saws of lengths 200 n 10000 and 200 n 3000 respectively

on the dominance of trivial knots among saps on a cubic lattice the knotting probability is defined by the probability with which an n step self avoiding polygon sap with a fixed type of knot appears in the configuration space we evaluate these probabilities for some knot types on a simple cubic lattice for the trivial knot we find that the knotting probability decays much slower for the sap on the cubic lattice than for continuum models of the sap as a function of n in particular the characteristic length of the trivial knot that corresponds to a half life of the knotting probability is estimated to be 2 5 105 on the cubic lattice

on two dimensional self avoiding random walks following nienhuis 1982 exact evaluation of the connective constant of the honeycomb lattice self avoiding walk model and the exact exponent values he has conjectured the author has re examined the available data on all three regular two dimensional lattices in the case of the triangular lattice the author has additionally corrected and extended the extant series the author finds support for nienhuis value gamma 111 32 for all three lattices and further finds that alpha 1 2 for all three lattices as given by nienhuis result nu 3 4 and the hyperscaling relation d nu 2 alpha the author is unable to find any consistent evidence of a correction to scaling exponent delta 1 1 cannot be ruled out

finding the maximizers of the information divergence from an exponential family this paper investigates maximizers of the information divergence from an exponential family e it is shown that the ri projection of a maximizer p to e is a convex combination of p and a probability measure p with disjoint support and the same value of the sufficient statistics a this observation can be used to transform the original problem of maximizing d e over the set of all probability measures into the maximization of a function dr over a convex subset of ker a the global maximizers of both problems correspond to each other furthermore finding all local maximizers of dr yields all local maximizers of d e this paper also proposes two algorithms to find the maximizers of dr and applies them to two examples where the maximizers of d e were not known before

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

asymptotics of knotted lattice polygons we use monte carlo methods to investigate the asymptotic behaviour of the number and mean square radius of gyration of polygons in the simple cubic lattice with fixed knot type let be the number of n edge polygons of a fixed knot type in the cubic lattice and let be the mean square radius of gyration of all the polygons counted by if we assume that where is the growth constant of polygons of knot type and is the entropic exponent of polygons of knot type then our numerical data are consistent with the relation where is the unknot and is the number of prime factors of the knot if we assume that then our data are consistent with both and being independent of these results support the claims made in janse van rensburg and whittington 1991a 24 3935 and orlandini et al 1996 29 l299 1998 topology and geometry in polymer science ima volumes in mathematics and its applications berlin springer