A method for systematic site-to-bond conversion of directed graphs

A graph-theoretic method for the complete derivability of bond graphs from their site counterparts is described. Bond perimeter, valence, and cyclomatic number distributions as well as spatial extent measures can be systematically generated when coupled with a site valence partition in the original graph. Relevant consequences for bond configurations include the facts that (i) percolation perimeter and cyclomatic number distributions are equivalent, (ii) geometrical susceptibilities are not independent, and (iii) a critical site/bond ratio exists.     

Finite lattice Z(3) models in two and three dimensions

We obtain the exact partition functions for finite lattice Z(3) models in two and three dimensions. We find the zeros of these partition functions in complex coupling constant space and interpret their positions in terms of the phase transitions of these models.     

Partition function zeros and the three-dimensional Ising spin glass

We have computed the exact partition function of the 3D Ising spin glass on lattices of effective size 3×3×L_z, 4×4×L_z, and 5×5×L_z forL _z up to 9, and several random bond configurations. Studying the distribution of zeros of the associated partition functions, we find further evidence that these systems have a singularity in the thermodynamic limit.     

Three Attractive Osculating Walkers and a Polymer Collapse Transition

Consider n interacting lock-step walkers in one dimension which start at the points {0,2,4,...,2(n?1)} and at each tick of a clock move unit distance to the left or right with the constraint that if two walkers land on the same site their next steps must be in the opposite direction so that crossing is avoided. When two walkers visit and then leave the same site an osculation is said to take place. The space-time paths of these walkers may be taken to represent the configurations of n fully directed polymer chains of length t embedded on a directed square lattice. If a weight λ is associated with each of the i osculations the partition function is $Z_t^{(n)} (\lambda ) = \sum\nolimits_{i = 0}^{\left\lfloor {\tfrac{{(n - 1)t}}{2}} \right\rfloor } {z_{t,i}^{(n)} } \lambda ^i $ where z ⁽ⁿ⁾ _t,i is the number of t-step configurations having i osculations. When λ=0 the partition function is asymptotically equal to the number of vicious walker star configurations for which an explicit formula is known. The asymptotics of such configurations was discussed by Fisher in his Boltzmann medal lecture. Also for n=2 the partition function for arbitrary λ is easily obtained by Fisher's necklace method. For n>2 and λ≠0 the only exact result so far is that of Guttmann and Vöge who obtained the generating function $G^{(n)} (\lambda ,u) \equiv \sum\nolimits_{t = 0}^\infty {Z_t^{(n)} (\lambda )u^t } $ for λ=1 and n=3. The main result of this paper is to extend their result to arbitrary λ. By fitting computer generated data it is conjectured that Z ⁽³⁾ _t (λ) satisfies a third order inhomogeneous difference equation with constant coefficients which is used to obtain $$G^{(n)} (\lambda ,u) = \frac{{(\lambda - 3)(\lambda + 2) - \lambda (12 - 5\lambda + \lambda ^2 )u - 2\lambda ^3 u^2 + 2(\lambda - 4)(\lambda ^2 u^2 - 1){\text{ }}c(2u)}}{{(\lambda - 2 - \lambda ^2 u)(\lambda - 1 - 4\lambda u - 4\lambda ^2 u^2 )}}$$ where $c(u) = \tfrac{{1 - \sqrt {1 - 4u} }}{{2u}}$ , the generating function for Catalan numbers. The nature of the collapse transition which occurs at λ=4 is discussed and extensions to higher values of n are considered. It is argued that the position of the collapse transition is independent of n.     

Small-scale properties of the KPZ equation and dynamical symmetry breaking

A functional integral technique is used to study the ultraviolet or short distance properties of the Kardar–Parisi–Zhang (KPZ) equation with white Gaussian noise. We apply this technique to calculate the one-loop effective potential for the KPZ equation. The effective potential is (at least) one-loop ultraviolet renormalizable in 1, 2, and 3 space dimensions, but non-renormalizable in 4 or higher space dimensions. This potential is intimately related to the probability distribution function (PDF) for the spacetime averaged field. For the restricted class of field configurations considered here, the KPZ equation exhibits dynamical symmetry breaking (DSB) via an analog of the Coleman–Weinberg mechanism in 1 and 2 space dimensions, but not in 3 space dimensions.     

M‐strings,elliptic genera and N = 4 string amplitudes

We study mass‐deformed N = 2 gauge theories from various points of view. Their partition functions can be computed via three dual approaches: firstly, (p,q)‐brane webs in type II string theory using Nekrasov's instanton calculus, secondly, the (refined) topological string using the topological vertex formalism and thirdly, M theory via the elliptic genus of certain M‐strings configurations. We argue for a large class of theories that these approaches yield the same gauge theory partition function which we study in detail. To make their modular properties more tangible, we consider a fourth approach by connecting the partition function to the equivariant elliptic genus of ℂ² through a (singular) theta‐transform. This form appears naturally as a specific class of one‐loop scattering amplitudes in type II string theory on T², which we calculate explicitly.     

Returns to the time axis in directed configurations

Directed configurations in dimensions 2 to 4 are investigated regarding the average number of returns to the anisotropy, or time, axis. Series expansions for this quantity, show that it vanishes asymptotically with a dimension dependent exponent equal to the critical exponent for directed animals in the corresponding dimensionality.     

Asymptotic mass degeneracies in conformal field theories

By applying a method of Hardy and Ramanujan to characters of rational conformal field theories, we find an asymptotic expansion for degeneracy of states in the limit of large mass which isexact for strings propagating in more than two uncompactified space-time dimensions. Moreover we explore how the rationality of the conformal theory is reflected in the degeneracy of states. We also consider the one loop partition function for strings, restricted to physical states, for arbitrary (irrational) conformal theories, and obtain an asymptotic expansion for it in the limit that the torus degenerates. This expansion depends only on the spectrum of (physical and unphysical) relevant operators in the theory. We see how rationality is consistent with the smoothness of mass degeneracies as a function of moduli.     

Symmetric vertex models on planar random graphs

We solve a 4-(bond)-vertex model on an ensemble of 3-regular (Φ³) planar random graphs, which has the effect of coupling the vertex model to 2D quantum gravity. The method of solution, by mapping onto an Ising model in field, is inspired by the solution by Wu et.al. of the regular lattice equivalent – a symmetric 8-vertex model on the honeycomb lattice, and also applies to higher valency bond vertex models on random graphs when the vertex weights depend only on bond numbers and not cyclic ordering (the so-called symmetric vertex models).The relations between the vertex weights and Ising model parameters in the 4-vertex model on Φ³ graphs turn out to be identical to those of the honeycomb lattice model, as is the form of the equation of the Ising critical locus for the vertex weights. A symmetry of the partition function under transformations of the vertex weights, which is fundamental to the solution in both cases, can be understood in the random graph case as a change of integration variable in the matrix integral used to define the model.Finally, we note that vertex models, such as that discussed in this paper, may have a role to play in the discretisation of Lorentzian metric quantum gravity in two dimensions.     

On the evaluation of a certain class of Feynman diagrams in x-space: Sunrise-type topologies at any loop order

We review recently developed new powerful techniques to compute a class of Feynman diagrams at any loop order, known as sunrise-type diagrams. These sunrise-type topologies have many important applications in many different fields of physics and we believe it to be timely to discuss their evaluation from a unified point of view. The method is based on the analysis of the diagrams directly in configuration space which, in the case of the sunrise-type diagrams and diagrams related to them, leads to enormous simplifications as compared to the traditional evaluation of loops in momentum space. We present explicit formulae for their analytical evaluation for arbitrary mass configurations and arbitrary dimensions at any loop order. We discuss several limiting cases in their kinematical regimes which are e.g. relevant for applications in HQET and NRQCD. We completely solve the problem of renormalization using simple formulae for the counterterms within dimensional regularization. An important application is the computation of the multi-particle phase space in D-dimensional space-time which we discuss. We present some examples of their numerical evaluation in the general case of D-dimensional space-time as well as in integer dimensions D = D₀ for different values of dimensions including the most important practical cases D₀ = 2, 3, 4. Substantial simplifications occur for odd integer space-time dimensions where the final results can be expressed in closed form through elementary functions. We discuss the use of recurrence relations naturally emerging in configuration space for the calculation of special series of integrals of the sunrise topology. We finally report on results for the computation of an extension of the basic sunrise topology, namely the spectacle topology and the topology where an irreducible loop is added.     

Global aspects of gauged Wess-Zumino-Witten models

A study of the gauged Wess-Zumino-Witten models is given focusing on the effect of topologically non-trivial configurations of gauge fields. A correlation function is expressed as an integral over a moduli space of holomorphic bundles with quasi-parabolic structure. Two actions of the fundamental group of the gauge group is defined: One on the space of gauge invariant local fields and the other on the moduli spaces. Applying these in the integral expression, we obtain a certain identity which relates correlation functions for configurations of different topologies. It gives an important information on the topological sum for the partition and correlation functions.     

Spin Foam Perturbation Theory for Three-Dimensional Quantum Gravity

We formulate the spin foam perturbation theory for three-dimensional Euclidean Quantum Gravity with a cosmological constant. We analyse the perturbative expansion of the partition function in the dilute-gas limit and we argue that the Baez conjecture stating that the number of possible distinct topological classes of perturbative configurations is finite for the set of all triangulations of a manifold is not true. However, the conjecture is true for a special class of triangulations which are based on subdivisions of certain 3-manifold cubulations. In this case we calculate the partition function and show that the dilute-gas correction vanishes for the simplest choice of the volume operator. By slightly modifying the dilute-gas limit, we obtain a nonvanishing correction which is related to the second order perturbative correction. By assuming that the dilute-gas limit coupling constant is a function of the cosmological constant, we obtain a value for the partition function which is independent of the choice of the volume operator. Member of the Mathematical Physics Group, University of Lisbon.     

The replica method and Toda lattice equations for QCD3

We consider the ε-regime of QCD in 3 dimensions. It is shown that the leading term of the effective partition function satisfies a set of Toda lattice equations, recursive in the number of flavors. Taking the replica limit of these Toda equations allows us to derive the microscopic spectral correlation functions for the QCD Dirac operator in 3 dimensions. For an even number of flavors we reproduce known results derived using other techniques. In the case of an odd number of flavors the theory has a severe sign problem, and we obtain previously unknown microscopic spectral correlation functions.     

Ground States of the 2D Sticky Disc Model: Fine Properties and N 3/4 Law for the Deviation from the Asymptotic Wulff Shape

We investigate ground state configurations for a general finite number N of particles of the Heitmann-Radin sticky disc pair potential model in two dimensions. Exact energy minimizers are shown to exhibit large microscopic fluctuations about the asymptotic Wulff shape which is a regular hexagon: There are arbitrarily large N with ground state configurations deviating from the nearest regular hexagon by a number of ～N ^3/4 particles. We also prove that for any N and any ground state configuration this deviation is bounded above by ～N ^3/4. As a consequence we obtain an exact scaling law for the fluctuations about the asymptotic Wulff shape. In particular, our results give a sharp rate of convergence to the limiting Wulff shape.     

Efficient algorithm for random-bond ising models in 2D

We present an efficient algorithm for calculating the properties of Ising models in two dimensions, directly in the spin basis, without the need for mapping to fermion or dimer models. The algorithm computes the partition function and correlation functions at a single temperature on any planar network of N Ising spins in O(N;{3/2}) time or less. The method can handle continuous or discrete bond disorder and is especially efficient in the case of bond or site dilution, where it executes in O(NlnN) time near the percolation threshold. We demonstrate its feasibility on the ferromagnetic Ising model and the +/-J random-bond Ising model and discuss the regime of applicability in cases of full frustration such as the Ising antiferromagnet on a triangular lattice.     

A transient testing technique for the determination of matrix parameters of acoustic systems,II: Experimental procedures and results

This paper deals with the application of a transient testing technique for the determination of matrix parameters of acoustic systems. Experiments were conducted for six systems of various geometrical configurations such as (1) a section of pipe, (2) a small expansion chamber, (3) a large expansion chamber, (4) partition pipes, (5) partition chambers and (6) an expansion chamber with insertion pipes. Two series of results were obtained for these systems, one concerned with the evaluation of matrix parameters for blockable and reciprocal cases and the other for the non-blockable and reciprocal types. Excellent agreement between experimental and theoretical results is illustrated by some typical results obtained for systems (1) and (3).     

String tension in SU(3) lattice gauge theory

Wilson loop expectation values have been determined in SU(3) lattice gauge theory without fermions using Monte Carlo methods and considering lattices of up to 10⁴ sites. A heat bath technique has been developed in order to enhance the statistical independence of successive lattice configurations.     

Modified Brans–Dicke theory of gravity from five-dimensional vacuum

We investigate, in the context of five-dimensional (5D) Brans–Dicke theory of gravity, the idea that macroscopic matter configurations can be generated from pure vacuum in five dimensions, an approach first proposed by Wesson and collaborators in the framework of 5D general relativity. We show that the 5D Brans–Dicke vacuum equations when reduced to four dimensions (4D) lead to a modified version of Brans–Dicke theory in 4D. As an application of the formalism, we obtain two 5D extensions of 4D O’Hanlon and Tupper vacuum solution and show that they lead two different cosmological scenarios in 4D.     

Ab initio calculation of strain effect on the magnetic properties of the thiogermanate [(CH3)4N]2FeGe4S10

The magnetic properties of the thiogermanate TMA(2)FeGe(4)S(10) (TMA=[(CH(3))(4)N](+)), and the influence of large strain are investigated by ab initio density functional theory calculations. An analysis of the electronic structure is provided considering antiferromagnetic (AFM), ferromagnetic (FM) and nonmagnetic configurations for the thiogermanate at the equilibrium states. A small difference in total energy between the FM and AFM states suggests that the thiogermanate TMA(2)FeGe(4)S(10) may possess a low Curie temperature. Changes in electronic structure and magnetic moment of the thiogermanate under strain are closely related to the distortion of FeS(4) tetrahedral units. With the help of a simplified molecule model, we show that, while the origin of the drastic change in magnetism under high isotropic compressions mainly originates from the decrease in the Fe-S interatomic bond length, the changes in the electronic structure between - 10% and + 15% isotropic strains are mainly due to the variations of the interatomic bond angles. Little effect of shear strain on the magnetic properties is found since the FeS(4) tetrahedral units rotate under shear, hence keeping their shape.     

Infinitely large new dimensions

We construct intersecting brane configurations in anit-de Sitter (AdS) space which localize gravity to the intersection region, generalizing the trapping of gravity to any number n of infinite extra dimensions. Since the 4D Planck scale M(Pl) is determined by the fundamental Planck scale M(*) and the AdS radius L via the familiar relation M(2)(Pl) approximately M(2+n)(*)L(n), we get two kinds of theories with TeV scale quantum gravity and submillimeter deviations from Newton's law. With M(*) approximately TeV and L approximately submillimeter, we recover the phenomenology of theories with large extra dimensions. Alternatively, if M(*) approximately L-1 approximately M(Pl), and our 3-brane is at a distance of approximately 100M(-1)(Pl) from the intersection, we obtain a theory with an exponential determination of the weak/Planck hierarchy.